Composition of Pretrained Diffusion Models: A Logic-Based Calculus
Authors: Peter Blohm
Vikas K Garg
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion
Score Composition
Logical Inference
Temperature Scaling
Fuzzy Logic
Primary area: generative models
TL;DR: We propose a set of novel score composition operators based on fuzzy logic for diffusion models and showcase that they improve combinatorial reasoning capabilities of ensembles.

Abstract: Composing pretrained diffusion models provides a cost-effective mechanism to encode constraints and unlock complex generative capabilities. Prior work relies on crafting compositional operators that seek to extend set-theoretic notions such as union and intersection to diffusion models, e.g., using a product or mixture of the underlying energy functions. We expose the inadequacy and inconsistency of combining these operators in terms of limited mode coverage, biased sampling, instability under negation queries, and failure to satisfy basic compositional laws such as idempotency and distributivity. We introduce a principled calculus grounded in fuzzy logic that resolves these issues. 
Specifically, we define a general class of conjunction, disjunction, and negation operators that generalize the classical mixtures, illustrating how they circumvent various pathologies and enable precise combinatorial reasoning with score models. Beyond existing methods, the proposed *Dombi* operators yield complex generative outcomes, such as the Exclusive-OR (XOR) of individual scores. We establish rigorous theoretical guarantees on the stability and temperature scaling of Dombi compositions, and derive Feynman-Kac correctors to mitigate the sampling bias in score composition. Empirical results on image generation with stable diffusion and multi-objective molecular generation substantiate the conceptual, theoretical, and methodological benefits. Overall, this work lays the foundation for systematic design, analysis, and deployment of diffusion ensembles.
Code is available at [https://github.com/Aalto-QuML/logic-diffusion-composition](https://github.com/Aalto-QuML/logic-diffusion-composition)

Published as a conference paper at ICLR 2026
COMPOSITION OF PRETRAINED DIFFUSION MODELS: A LOGIC-BASED CALCULUS
Peter Blohm
Vikas Garg
Aalto University peter.blot www.hhhaalat..o....o
Aalto University and Yai Ltd vga rg@csail m.m.t.t....... mit.cc edu
ABSTRACT
Composing pretrained diffusion models provides a cost-effective mechanism encode constraints and unlock complex generative capabilities, Prior work re- lies on crafting compositional operators that seek to extend set-theoretic notions such as union and intersection to diffusion models, e.g., using product mix- ture the underlying energy functions. We expose the inadequacy and incon- sistency of combining these operators in terms of limited mode coverage, biased sampling, instability under negation queries, and failure to satisfy basic compo- sitional laws such as idempotency and distributivity. We introduce a principled calculus grounded in fuzzy logic that resolves these issues. Specifically, we define general class of conjunction, disjunction, and negation operators that general- ize the classical mixtures, illustrating how they circumvent various pathologies and enable precise combinatorial reasoning with score models. Beyond exist- ing methods, the proposed Dombi operators yield complex generative outcomes, such as the Exclusive-OR (XOR) of individual scores. We establish rigorous theoretical guarantees the stability and temperature scaling of Dombi com- positions, and derive Feynman-Kac correctors to mitigate the sampling bias score composition. Empirical results image generation with stable diffusion and multi-objective molecular generation substantiate the conceptual, theoretical, and methodological benefits. Overall, this work lays the foundation for system- atic design, analysis, and deployment of diffusion ensembles. Code is available at g i thub. com / Aa tto--oott logi c-di fusii on compos i iion.
INTRODUCTION
Pretrained general-purpose generative machine learning models (Devlin et al., 2019; Brown et al., 2020) have become practically synonymous with the term artificial intelligence itself. Their vast capabilities (Bommasani, 2021; Wei et al., 2022), however, come at the cost of excessive need for growing datasets (Kaplan et al., 2020; Villalobos et al., 2022), and yet additional techniques needed to reach adequate performance in downstream tasks. Finetuning (Devlin et al., 2019), human-feedback- based reinforcement learning (Christiano et al., 2017; Ouyang et al., 2022; Zhang al., 2023), retrieval augmented generation (Lewis al.. 2020), or even specialized prompting techniques (Brown et al., 2020) are then used to retrofit models to specialized tasks and domains.
As alternative monolithic general models, compositional generation (Jordan & Jacobs, 1994; Hinton, 1999; 2002; Yuksel et al., 2012; Vedantam et al., 2018; Du et al., 2020) seeks to combine the domain knowledge from different models or different guidance signals- to solve a task at hand. As many models follow probabilistic formulations, using probabilistic language for composition is a natural approach. Products of Experts (PoEs) (Hinton, 1999; 2002; Liu et al., 2022; Du et al., 2023; Skreta et al., 2025a) have been devised and widely used as mechanism enforce conjunc- tive constraints, with the idea that their product is only large when all components are large. The assumption underlying this approach to model joint distributions statistical factors however, does not in general hold.
distributiions-stttisiiiallll independence of the
Often tackled separate problem is the concept avoidance in generation. Similar other tasks, unlearning (Ginart et al., 2019: Nguyen et al., 2022; Wang et al., 2024) as a specific form of fine- tuning or post-training avoidance and steering methods (Dhariwal & Nichol, 2021; Ho & Salimans,

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Failure Modes in Prob. Operators
Dombi Operators
FKC for Sampling
ubb
II
x
Openarr Dijuucttion Cosjunction
(w)())
sxx)
Detinition and Algebraic Properties
Inconsistent Scaling
Tradeoffs for different choices of A
Experiments
dale dt Dynamic Stability
& (Arl)
p Algebraic Stability
Negation Instability
Figure 1: Overview of Paper Structure and Main Contributions.
2021; Dong et al., 2023; Garipov et al., 2023; Kirchhof et al., 2025) have been proposed, which often utilize PoE with inverse probability densities for avoidance rely on training additional models.
In this paper, we investigate the composition of diffusion models from fuzzy set-theoretic and fuzzy-logic perspective. We propose a procedure derive of well-behaved composition opera- tors, and among them, propose Dombi operators in Section 4 as a one-parameter family, extending and uniting commonly used operations such as mixture of experts (Jordan & Jacobs, 1994) (MoE), harmonic mean (Garipov et al., 2023) (HM), and as special case, the geometric mean a tempered Product of Experts (Hinton, 1999) (PoE), as visualized in Figure fective methods, our approach is purely online and utilizes pre-trained diffusion models and online density estimates, An overview of our main contributions is provided Figures and 6.
contrast to many existing
BACKGROUND AND RELATED WORK
2.1 SCORE-BASSEL MODELS
We want approximate probability distribution defined over Rdd to sample from it. In the context of score-based modelling (Song et al., 2021). we first recast p as a Boltzmann distribution, and let the model learn the score funcctinn sæ (x) s V p(x),, avoiding the unknown partition function. To facilitate sampling via MCMC, the data distribution p is gradually destroyed according to the forward noising SDE (Øksendal, 2003)
dx, = Jr (x)d7 d.dWr, ~ p(xo)-
Here fr Rd + R is some, usually linear, drift function and ar R + is time-dependent diffusion coefficient and W, the Wiener process. These functions are chosen such that x,=1 (0, Ia). the standard Gaussian. For sampling, we simulate the backward process with t = 1 7 as
dx; = fx(exx + of Vx (x:) dt + ordw..
(1)
which satisfies the Fokker- Planck- equation
0p.(x) -V
V log)((x))) + of Ap,(x).
Øt
2
where Apt denotes the Laplacian of Pt and (V,.) is the divergence operator. For the rest of this paper, we assume we are given a set of pre-trained score models {st) which model the respective probability distributions {pi]e For statements about the f = 1, we omit the index.
To translate the theory developed in this paper to practice, rely efficient density estimation to assign responsibility to score functions. We can efficiently estimate densities during inference with Itô's Lemma (Karczewski et al., 2025a; Skreta et al., 2025b)
o? dloopt z =( (dxi, sxx)) + (IV, f((x)) + (f(())), sx()))
2 [|s((x))1²)
(3)
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2.2 COMPOSITION OF SCORE FIELDS
There is a quickly growing body of work on compositions, mixtures, and products of energy-based models (EBMs), well flow and diffusion models. We explicitly focus on training-free mixtures of score functions in diffusion. Prior work (Du et al., 2020; Ho & Salimans, 2021; Skreta et al., 2025a;b; Gaudi et al., 2025) mainly bases composition on probabilistic operations on the underlying distributions. As the interpretation of these operations is often logical or set-theoretic, we will use the symbols {V, A, - to denote them, for both probability densities and their scores. In score-based modelling, conjunctions are then usually represented by (sometimes geometric) products
(x) Ax (xx) == p² (x))²(x) + s (x) Ax (x) s² (x) + (x()
(4)
(x())) and disjunctions by mixtures, where we the weighting a² +x=))* with
1 5p' 2x)x)) ()) p (x) V+ px(x) =
(5) Two noteworthy exceptions from product-based conjunctions are Garipov al. (2023), who model conjunctions with the harmonic mean p² (x)p²(x), (x'(x) + p²())) and Skreta et al. (2025b), who reweigh individual scores steer towards equal density directly.
- (x) V+ (x) = a ss (x) + x²2²(x))
Importantly, under the usual dynamics of diffusion processes, for t + 1, nonlinear compositions do not commute with the noising operator, i.e., pd V+ P = (p' V+ p²); but P! Ax p # (p' Ax p²): This means that naive composition of perturbed score models leads to bias that can be corrected with methods like sequential Monte Carlo (SMC) (Skreta et al., 2025a; Thornton et al., 2025). The typical formulation of Equations (I) and (2) is then extended to weighted SDEs, where samples have time-dependent log-weights wy which are defined via the weight field gx) as
dw: = [ )dl. = @p)(x) = J. (x)p((xx), with F(x) = ge (x)
/ gr (x)pe(xx))
Øt
These weighted SDES with (x) then must satisfy the Feynman-Ka Kac PDE
Op:(x) (V, oiV logp((x))) + -
Ap.(x) + (x)pe((xx.
(6)
For nonlinear score operations like annealing, CFG, or PoE, Skreta et al. (2025a) then explicitly derive the biases incurred by approximating the true composed distribution with the composition of noisy scores, collect the "left-over" terms in g. and use additional correction methods. We adapt their formalism to improve the simulation of our operators in Section 4.
To avoid certain distributions, EBM's and score models are usually only negated relative to oth- ers (Vedantam al. 2018; Du et al., 2020; 2023; Garipov et al., 2023; Dong et al., 2023; Skreta et al., 2025a; Gaudi et al., 2025), as also done in classifier-fres guidance (Ho & Salimans, 2021) (CFG). In these settings, independent concept negation (ICN) for concept y is often defined, for 0 < 7 1 in the EBM context (Hinton, 2002; Du & Kaelbling, 2024). In more recent work (Liu et al., 2022: Du et al., 2020; Ho & Salimans, 2021), often the formulation p(x| -v) x ((x))+)((xl))" used instead, derived via Bayes rule.
as p(x p(x)/p(xlx))
From a perspective of logic, these variants make use of the reciprocal as pseudo-inverse -p(xly) 1/p(xly), but to our knowledge, explicit negations in score-models are not often explored or theo- retically justified, and alternatives (Chang et al., 2024) also lack clear theoretical interpretation.
2.3 FuzzY LOGIC
Our proposed method directly draws from the theory of fuzzy logic. Fuzzy logic relaxes classi- cal logic from a binary domain to real-valued memberships and notation from Klement et al. (2013) for the following concepts. We define t-norm, gen- cralization of conjunction or intersection operations, as T : [0, 1]2 - [0, 1] which is commuta- tive, associative, monotonously increasing in either variable, and fulfills the boundary condition Vx E [0. 1] : T(x, R 1) x. With the standard negation N(x) = x, we define the dual t-conorm S: [0, 1/2 + J0, 1] of T, disjunction, via De Morgan's 's law as S(x, y) = (2))
[0, 1). We follow the definitions
T-norms that are strict, i.c., continuous and strictly increasing, can be generated (Dombi, 1982; Klement et al., 2013) by continuous, strictly decreasing function f [0, 1] + [0, oc] with f(1) = 0.
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so-called additive generator, i.e., [(x, = ff + f(y)) For this work, the parametrised
Dombi t-norm is the most important representative, generated by fx(z) = ( - 1)^ A favorable property of the Dombi t-norm is that limx-xoo Tx Tas min. The min t-norm Tar together with SM = max is the only continuous De Morgan dual that is idempotent with Ta (2r, = x and distributive with (x, Sas =)) ( y), (x,, (Klement et al., 2013). To make the domain of probability densities compatible with the theory of fuzzy logic, we utilize some bijective, order-preserving function Φ [oo) + [0, 1] which converts densities into fuzzy membership.
FAILURE MODES IN SCORE COMPOSITION
We provide further motivation for our approach with brief illustration of the mismatch between expectations and actual behaviour score composition when using PoE and MoE as operators. We illustrate clear stability concern when avoiding concepts, well unintended biases in the sampling dynamic caused by score rescaling. Finally, we demonstrate that interpreting PoE and MoE as set operators easily leads to modelling mistakes.
3.1 NEGATION INSTABILITY
The two distinct EBM-style negation px(x)/p²(x) and CFG-style negation p pxx) (x" operators both easily translate score calculus. However, specifically, the EBM-style negation shifts the target distribution and requires careful calibration of to avoid exploding tails (Garipov al., 2023; Chang al., 2024: Ban et al., 2024). The commonly used CFG-style negation behaves much more stably in practice, yet theoretical arguments for its use remain limited. While better behaved, CFG negation might overaccentuate regions in the target distribution p² where p² vanishes (Chidambaram et al., 2024). In conclusion, neither version of this operator guarantees normalizable distribution and both tend to exhibit unwanted tail behaviour as depicted in Figure 2a. We revisit this issue in Section 4.2.
3.2 INCONSISTENT TEMPERATURE SCALING
PoE uses the additive score calculus s Ax 82 = 32 This leads to a scaling of scores depending on their alignment: ||s Ax 3 +| 2 + ||| cos 0, where is the angle between s s². In diffusion, temperature main methods control the behavior of the model (Guo et al., 2017: Chidambaram et al., 2024; Karezewski et al., 2025a:b). Crucially, the score alignment (0) may vary significantly depending on location and the models used.
the score vectors-is is one of the
In regions with high score alignment (small 0), temperature is decreased, and the composition is bi- ased towards higher density regions than what dictated by any component. Conversely, in regions with low alignment between scores (large 9), the temperature is increased, discouraging higher density regions. Figure 2b illustrates this behavior in contrast the Dombi operators, which guar- antee ||s Ax s²1l < max(]]s I|. ||> This problem persists when scaling with, e.g., the geometric mean or score averaging s (x)/2 2 + s² (x)/2, where the score norm still varies across positions.
3.3 COMPOSITION PROPERTIES
Model composition is often interpreted as logicul/set operation over the hypothetical sets of "typi- cal" samples of each distribution. This interpretation leads to pitfalls with probabilistic operators, as MoE and PoE do not exhibit the necessary algebraic properties expected of logical or set operators. simple example of this is avoiding multiple distributions p" and pz individually. Intuitively, one might use conjunction over multiple negated distributions. The resulting operation, however, does not match the expected result, as negations and products commute:
p' 1T
P
pd Ax 1 p
=
Ax p²) +
p²
Ax
(p
V p3)
1 +
In general sense, PoE is also neither idempotent, p Ab one direction, i.c., (p' ^x p²) V pd + (p' V "²) ^x (p² V V+p²), This severely restricts the options for rewriting compositions for different purposes, such as collecting terms.
p=p²
# p and distributes only in
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Density
Density
Prob: plq/
p
Prob: pqr
Dombi: рdλ
Dombi: p A 2r
(a) Avoiding g: density ratios vs. Dombi.
(b) Intersection: products Dombi.
Figure 2: Failure Modes of Composition with probabilistic approaches. (a) illustrates that density ratios can lead to unstable behaviour, compared to the bounded referenced Dombi negation. (b) shows an intersection, where the product can lead to locally underscaled and overscaled tempera- tures, contrast to Dombi composition.
DOMBI OPERATORS
In this section, we present our main contribution: a principled score-calculus based on the Dombi I-norm Dombi (1982). We present a more general definition and derivation of class of De Morgan dual operators for scores and densities in Appendix B.
We extend the Dombi t-norm, generated by fA(x) = - 1)) for X Ry0, and lift it from fuzzy memberships to densities a semantically meaningful way. This is achieved by the referenced map- ping function &e that expresses negation with reference some (pointwise) constant function c(x). This constant can be interpreted as a normalising factor and serves as a fixed point for negations, but does affect other operations. In the context of distributions, this normalization by reference distribution c(x) is analogous to the probability ratios used in CFG, or the PoE conjunction, e.g.. pre-
sented by Liu al. (2022). With abuse of notation, will write (p(x)) (.(p:) pxx)+c(x)-
p(x)
Definition 4.1 (Dombi Operators). Choose À R>o and a continuously differentiable function p(x)
Rd > R>o with se Vxloge. For fx(x) 1) and Φc(p(x)) p(x)+((x)) let a
exp(A (x)) Es)(1 exp(A (x)] The Dombi operatirs are the De Morgan dual operators induced by Sx. De:
-p(x) = c(x)2
sx() = -2)((x) - -s(x)
p(x)
(xx))x²(xx)
(xx) = (x)x + 1² (x))))1
s*(x) ^ 52 a² (x)==(() + 0² xx)
(8)
vxx -x(x)) (x) Vv 2x of (x() + 2
(9)
A detailed derivation of this result can be found in Appendix B.3.
This definition bears multiple remarkable properties. As De Morgan dual operators, Dombi operators have favorable algebraic properties, described Proposition B.2. Most notably, Dombi operators approximate a Lattice, meaning they (approximately) adhere to the axioms of set opera- tions. In addition their algebraic properties, we can exactly recover established score operators.
4.1 RELATION OF DOMBI OPERATORS TO PROBABILISTIC DENSITY OPERATIONS
Dombi compositions over distributions are power norms, and with different choices for the exponent X, we recover well-known operators (illustrated in Figure 3), such as min/max for À - oc, linear mixturr/harmonic mean for À 1. For À 0. Dombi composition is undefined the score calculus for - 0 equivalent to the geometric mean. Most notably, this parametrization shows which operators work well calculus, visualized in Figure 3. The Dombi score calculus
densities, yet
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min
HM
PoE
MoE
max
Density gap (log Pl
Mixture Weight (a1)
Conjunction
Dual Operator
Disjunction
0.5
-00
-1
+00
Composition Parameter (2)
Figure 3: Visualisation of Dombi Composition px(x) о사 (x()), depending their density gap. Higher values À result in more sensitive weights a, and flipping the sign of À gives the De Morgan dual operator. Different choices of correspond known operators.
also equivalent to calculus induced by power means (Amari, 2007), which differ from the Dombi density operators by a constant factor of 1/2*. Power means are minimizers corresponding a- divergences such as the KL-divergence- between sets of distributions. This connection suggests further parallels of score calculus to work on logical operators via divergences (Khalafi et al., 2025).
4.2 PROPERTIES OF REFERENCED NEGATION
Under our definition, referenced negation yields an expression equivalent to CFG-style negation for y 1. A crucial distinction between the Dombi calculus and vanilla CFG arises when negation is combined with conjunction. We know that a negated distribution is normalizable if the x² divergence bounded. We then have, per definition (Nishiyama & Sason, 2020)
= =f (p(x) 9(x))2 gxx)
p(x)2 q(x)
2 (pllg)
dx ttr=
< 0o.
(10)
In practice, bounded x2 -iivergeeece cannot be easily verified, but we can guardrail against diverging distributions with Dombi conjunctions. Dombi conjunctions are upper bounded by min, single stable distribution prevents diverging composition results, regardless of the tail behaviour of other components. This illustrated in Figure 2a, and justifies the use of unrestricted CFG px, c(x) /p(x)" This non-involutive negation variant not involutive, but allows adjusting negative guidance behaviour-withor the risk of divergence.
5 INFLUENCE OF À ON DISTRIBUTIVITY AND MIXTURE STABILITY
Besides the connection prior work, the parameter À from the Dombi operators naturally appears as inverse temperature in the score composition. For À - oo, the Dombi operators recover the exact {min, max] lattice and with it distributive and idempotent behavior. For finite X, the simple bounds in Proposition B.5 can be used to quantify biases in density compositions. We use this to present a simple bound for the maximal density bias we introduce when applying distributive laws.
Corollary 5.1 (Idempotency and Distributivity Bias). Let Axi Vx be the Dombi density operators. From Proposition B.5 it follows that, for B Rzo B 2=2/ = 12 2/-2/2 2/2 B)
Yr E R200
I = 21/2,,
I EAx = 1/2
(11)
Vx, 3, = E R20:
I V E (( A (x VA x)) 22/A
(12)
Yz, Vr, E R
I F 2) ((* (x -2)))++))))) 22/A
These easily obtainable bounds trivially generalize to sharp bounds arbitrary compositions, allow- ing us to make immediate statements about the stability our composition. As our score coefficients vary during the inference process, would naturally be interested in the rate of change of these efficients, as drastic change rates might cause the composite model to "oscillate" between two scores, especially in conjunctions. As before, the statement can be extended more complex formulas.

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Proposition 5.2 (Mixture Stability). Let a2 softmax, (Alogp) Alog ). dombi composition p' x p². Then it holds for the score difference As = 87
Elda
xz]1 O(I - ||o]lAs||2ds
d(a}. = of
Together, Corollary 5.1 and proposition 5.2 quantify the tradeoff between compositional precision and mixture stability. High À results in small biases over the ground truth of the composition, but for large differences between the component scores ||s sEll. the mixing coefficients a' might drastically oscillate. When À is chosen smaller, the volatility of the mixture is naturally bounded.
6 PRECISE SAMPLING WITH FEYNMAN-KAG CORRECTION
While Definition 4.1 explicitly states how the densities and consequently the scores of our target distribution look, simulation with, e.g., dx; = fx,) + =f
(x) Λx 82(x)) dt ftt+-a &,dd will not sample from the desired marginals during the reverse process and consequently not from the correct target distribution pı (x) Ax px)).. Skreta et al. (2025a) introduce Feymman- Kac Correctors (FKCs) for diffusion, which correct for the biases of score composition. We recast the composition with Dombi operators as weighted SDEs, then collect all terms that are missing from our score proposal into the weight field g. At inference time, SMC methods like systematic sampling can be used to correct for these biases.
In this section, we extend the FKC terms to our Dombi operators, and refer to Appendix C.I for proofs. As the combi-compoostion just reduces to "power norms" our densities, as well as spe- cial case of geometric averages in the case of referenced negation, we present these two correction terms here. More complex compositions then propagate the weight-fields gr ge(x) of components.
Proposition 6.1 (Referenced Negation CFG+FKC, Skreta et al., 2025a). Consider two diffusion models q: (x), of (x) defined via the Feynman- Kaaman--Kac equation in Equation (6) with weights gf
The weighted SDE corresponding the referenced negation of Pe(x) x 7o (x) is, with dx;(x)= = (x)xdt
dx; = [xx) + of log o (xx) - V log de (x:))] dt + o,dw; (x) = V log qf (x,) - V log q.t (x;) + 29 (x) gt (x).
(13)
As stated in Equation (10), Pr (x) is then a normalizable probability distribution, if and only if ( ll < cc. We might also want anneal q² tune the "narrowness" of the concept we avoid. We propose combined annealing of the form (x(x)*+* /////// (x)" to allow tuning the two distribu- tions in relation to each other, while still maintaining slightly improved normalizability compared to the standard CFG, and maintaining an unbiased energy estimate for further composition.
Next, we state how FKC terms propagate through connectives. As both our connectives are essen- tially power-norms with positive negative exponent, both cases can be handled at once.
Theorem 6.2. Consider two weighted diffusion models qf (x). q (x) defined via the Feynman- Kac equation with weights ot (x), o² (x), and a parameter X E RR' [0]. The weighted SDE corresponding
to (x) x (et (x)* A + o7 (x) 1/2 with of [[(x]* (x)x E (0, 1), and dw; = (e is
(x(x)*
dx, (x)) + of (a V log of (xx) + af log oi (x di + a.dw,
sx(x) (1 2) 0² o²
Σ (E(1)2)
aj VVllg
Σ
af ox
+ Σ
(x))
sE(1.2)
(E(1,2)
(14)
Proposition 6.1 and theorem 6.2 are presented in a modular form. This allows us to use arbitrary combinations of operators and propagate the log-weights of components.
6.1 INFERENCE PROCEDURE
Together, Definition 4.1, proposition 6.1, and thcorem 6.2 define our theoretical basis for arbitrarily nested model composition. During the sampling process, keep track of the evolution of
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7
0
2
0
3 1 3 C
/ 2 0
0 2
2
0 Q
2
0 1 1
2
0
0
Mounluin Landscupe
Silhouetle
(a) 2 0 0
Ph
(b)
Pu 3
0 0
2
0
3 1 3
0 0
1
0 1
8
0
3
2
0
2
3
Psor ps
(d) Pxar -P8
Starry Night, Van Gugh
NOT Van Gogh
Figure 4: Generated Image Compositions with MNIST (A E {5 10 3,5 5 (0- and Stable
Diffusion (A 10).
loglikelihoods with the density estimator from Equation (3). This efficient density estimation method enables us perform complex model compositions with minimal overhead. During composition, we can then compose our scores, lig-liielssoods, and FKC terms with the procedure described in Algorithm 1. To improve sampling, trajectories (Naesseth et al., 2019). In our experiments, we use systematic sampling proportional to the exponentially weighted momentary weight-field (xp(a(()dd (Douc & Cappé, 2005).
SMC techniques during the simulation
Algorithm 1: COMPSSOONNN over arbitrary formulas
Input sccores {s') ) log-likelihoods (log K= weights (g') formula F il
Output: Composite score s. Composite log- lel--iikiihodd log q. Composite weight g
1 F then return s" .log q'
2 else if F 75 then return 2s - s 2log d' log of - s + 2g - // Frop. 6.1
3 else if F Fi Aa F3 then
Conjunction is negative power norm
/x Case
= F. AA F. RVA.:: evaluate subformulas first
4 ssl logg
DOMBICOMPOSITION (s'),11, (logg'}/-n: (g')... F))
5 s2. logg + DOMBICOMPOSITION( (s') (logg') 1 (g' 1. Fb)
6 a² softmax: (Alog Alogy") ); a² + 1-aa
7 Ø + (1 A) |100'8 + 2122 (a (a-11==112 a² 11135993)) // Theorem 6.2
return a²3s + 2²
LogSumExp(AAogg' Alogq' 2 T g + a²
EXPERIMENTS
7.1 COMBINATORIAL BIAS IN COMPOSITION SAMPLES
We first test the ability of our method to sample from complex compositions of diffusion models qualitatively and quantitatively. We compose three pretrained models that generate colored MNIST digits (LeCun, 1998). Our three models are defined
follows: Model Pı generates the digits {0. 1, in cyan, P2 generates digits smaller 2: {0, 1, 03 in cyan or beige and Ps generates the digits {0, 2} in cyan beige. We would now like to perform set operations
these 7 unique digits-color pairs, similar to Garipov et al. (2023), but with general operations. Figure shows set of chosen set operations models. Beyond the intersection pn p1 Ap2/pp and the
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Maja
XOR2
OneHot2
Ma
XOR10
OneHot to
Method
Satt
Unift
Sart
Unift
Sart
Uniff
Satt
Unift
Satt
Unift
Sart
Unift
Dombi
1.00
1.00
0.97
1.00
0.97
1.00
1.00
0.98
0.91
0.98
0.09
0.99
PoE/MoE
1.00
0.82
1.00
1.00
1.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
Table 1: Quantitative SAT--xperiment, with 2 samples from a composition of i models. We report Sat (fraction of particles around satisfying modes), and Unif, the normalized perplexity over satisfying modes (1 uniform; 0 collapsed). Higher is better for both metrics.
union pu = p1 Vp2 Vps we show results for the exclusive-or operation Pxor = (p) (p:) P- that samples digits from either p1 p2 but not from their intersection. We then show Pxor A P3 {2, 0} as well as Pxor A -ps = {3, }. In all experiments, the reference model for negation is chosen to be the composite score pu- With few exceptions, we can see that our approach allows us to sample from complex compositions such as Pxor solely via score composition of pretrained diffusion models.
Quantitative Results from a SAT-like toy experiment are shown in Table explained in detail in Appendix E. We model three different composition formulas over k = (2, ....... 10} models: Ma ig as a majority, XOR2 as formula with an exponential number of operators, One Hot,, as formula with many negated terms. Compared to PoE/MoE composition, our Dombi composition exhibits less mode-bias (higher Unif.), and is stable (higher SAT.) even in regimes with many negations. We emphasize that Dombi maintains Sat 0.9 even for XOR1o, which consists of over 1000 score terms.
7.2 MUL.TI-PROMPT IMAGE GENERATION AND AVOIDANCE
To show the performance of Dombi composition in production scale diffusion models, we compare its ability to generate images that interpolate between or avoid concepts using Stable Diffusion (SD) v1-4, For all our compositions, we choose two prompts C1, C22 e.g., "a mountain landscape and "a si silhouette of dog". " We then evaluate twenty pairs of images composed conjunc- tively, as p(x[c1) A p(x[c2). and compare against and by Skreta et al. (2025b) and scaled PoE, i.e. unweighted averaging of scores (Liu et al., 2022). We further investigate p(x (1) A (x(x))( on ten pairs of prompts to illustrate the ability of our model to avoid concepts. As baselines for con- trastive prompting, we use ICN (Ho & Salimans, 2021) and the conjunction of (Skreta et al., 2025b), combined with our referenced negation. We use the composed scores in the usual CFG pipeline of SD and measure for all prompts the min. CLIP score (Radford et al., 2021), which measures cosine similarity between image embedding and prompt embedding, and the minimum ImageReward value (Xu al., 2023), which estimates how closely generated images align with human preferences. For contrastive prompts, we report the difference of each metric between cl and c2.
Dombi Composition shows improvement beyond state-of-the-art methods in both CLIP and Im- ageReward scores, as shown in Tables and 5b, with an example of generated images in Figure 4. For the full list of used prompts, see Appendix D.2. A stark contrast between our method and SuperDiff is evident in Figure 5c, which shows mixture stability over the first 100 iterations of the generation process. The batch variances of the mixture coefficient with X, with an increase over time caused by different equilibrium points per batch. and (Superdiff) shows strong fluctuations in mixing coefficients, especially during the initial iterations. This effect is more pronounced when we retrofit and to contrastive settings with our negation definition.
shown to correlate strongly
7.3 MULTI-TARGET PROTEIN SYNTHESIS WITH FKC CORRECTION
As final experiment, we test Dombi composition combined with FKC in the setting of structure- based drug design (SBDD). The goal here is to generate molecules (ligands) using the structure of protein guide and evaluate their binding energy (Anderson, 2003). In our experiments, we investigate the impact of FKC from Theorem 6.2 on the quality of Dombi composed results. We generated 32 ligands of sizes {15, 19, 23, 27, 35} each, for 14 protein pairs, and evaluated their docking scores using Autodock Vina (Eberhardt et al., 2021) and reproduced the experimental setup of (Skreta et al., 2025a). In this experiment, we use annealing the base distributions: We evaluate
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CLIP (1)
Im. reward (1)
CL)
Im. reward (t)
Supertitt
Domb Domb
Mothod
A
mean std
incan std
Method
mcan_std
mean std
035
and
24.87
-1.33
ICN
2.207
1.14
PoE
24.41 24.41
-1.5520.70
and
5.44
0.61
0.15
Prompt
23.79 2.772
-1.61_0.75
0.1
25.25 1:2
-1.18 10.84
6.91
20.75
0.05
Dombi
25.32 65
1...6
Dombi
7.40 42200
1.11
100
10.0
25.50 22.64
-1.18
+0.86
10
7.02+2.64 +2.54
1.21 1.21+
Ioration
Improvementy haseline (and)
Impremements us baseline (1CN)
0.40
SuperbLff D
Domb
and
0.00 20.00
0.00
ICN
0.00
0.00
Pol
-0.46 10:31
-0.22
11.00
and
-1.85 23.54
+0.63 20777
Prompt
-5.11 =3.25
0.13
0.1
0.38 -2.67
0.15-0.56
-0.38 12.16
-0.24-o.a0
Dombi
0.46 62
0.18 20.80
Dombi
0.10 1..8
-0.03
20.63
10.0
0.64
0.16g0..2
10
-0.27
0.07 10.55
(a) Conjunctive prompts
(b) Contrastive prompts (A = 1)
Iterattio
(c) Mixture Stability
Figure 5: Joint generation performance with Stable Diffusion, and paired improvement over base- lines with 20 seeds. a shows results for 20 joint prompts p(x(er) A pxx(c2). ). shows results for 10
contrastive prompts p(x(c) (ei) A "p(x) (02) shows the variance of a during conjunctive (top)
and contrastive (bottom) composition. and is from SuferDiff (Skreta et al., 2025b).
Method
Temp y FKC? [F, Pt)
max(P,, (C)
Better than ref. (1)
Div. (1)
Val. Uniq. (t)
QED (f)
SA (
TagetDiff
62:19 27.66
0:32 0.0
0.89
0.86 our
0.57 4:144
0.59 0.00
Dombi
1
x
G8 60 2000
-7.42
0:28 0.44
0.88
0.96 4:000
0.58 eass
0.59
Dombi
1
-7.71
Dombi
2
71.36 2414
7.50
0.30 nm
0.88
0.93
0.59
0.62
0.00
Dombi
2
81.63
-8.55
0.38
0.55
0.93
0.59
0.62
Table 2: Docking Scores of generated ligands for 14 protein target pairs (P P2), in batches of 32
ligands for 5 molecule lengths each. We compare conjunction with Dombi (A = 1) with and without FKC with annealed base distribution and also report TargetDiff' from (Guan et al., 2023) as baseline.
p(x(P1)) A p(x(P2)7, and propagate the FKC term of the annealed base distributions to our dombi operator as in Algorithm Per batch, we report the average joint docking performance to each target protein as their product (21 * Pz), the objective of PoE. as well as max(P1, P2), which is closer to the objective of the Dombi composition. Further, we measure the fraction of molecules that have higher docking score than the known reference molecules, the diversity of molecules, as well as the fraction of valid and unique molecules, and their drug-likeness (QED) (Bickerton al., 2012) and how easy they are to synthesize (SA) (Ertl & Schuffenhauer, 2009).
FKC Correction improves the docking performance in annealed and unannealed settings, as shown in Table 2. The difference is more pronounced for 7 = 2, where we also collect FKC terms for the annealed base distributions. In Appendix D.3 we show results with an additional small sweep over À values, where the performance for À = 0.3 and X = 3 shows to be similar.
8 CONCLUSION AND FUTURE WORK
In this work, we introduced Dombi composition operators as a purely online, well-defined, general class of coopescooooi operators. Based on power norms, our method recovers and unifies prior work, like MoE, the harmonic mean, or contrast operators (Garipov et al., 2023), yet offers theo- retical benefits that are crucial to ensure stability when score compositions become more complex. An important future direction are dynamic schedules for X, as Proposition 5.2 suggests that adaptive choices depending on the scores might be better suited to ensure stability. This work opens up some exciting possibilities, e.g., potential applications in neurosymbolic methods, where modular diffu- sion models could be coupled to solve combinatorial tasks. Furthermore, the option to rewrite formu- las might be utilized to switch to different sampling techniques for, e.g., factoring out subformulas.
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REPRODUCIBILITY STATEMENT
Detailed proofs are provided in the Appendix for our theoretical results. We also provide a link GitHub repository containing all the code used reproduce the results in this manuscript', The Repository contains the details required to reproduce the empirical results, including our hyperparameter settings. We make our code public under the MIT License.
ETHICS STATEMENT
We provide here principled framework for composing pretrained diffusion models. Diffusion models can be composed to enable generative capabilities beyond what the individual models can provide. While the harmful effects due to misuse of composition cannot be ruled out, our work does not raise further specific ethical concerns that need to be highlighted here.
ACKNOWLEDGMENTS
VG acknowledges the support from the Research Council of Finland (grant 342077), Saab- WASP (grant 411025), and the Jane and Aatos Erkko Foundation (grant 7001703).
REFERENCES
Shun-ichi Amari Integration of stochastic models by minimizing a-divergence. Neural computa- lion, 19(10)2770-2776, 2007.
Amy C Anderson. The process of structure-based drug design. Chemistry & biology, 10(9):787- 797, 2003.
Yuanhao Ban, Ruochen Wang, Tianyi Zhou, Minhao Cheng, Boqing Gong, and Cho-Jui Hsieh. Understanding the impact of negative prompts: When and how do they take effect? In european conference on computer vision, pp. 190-206. Springer, 2024.
G Richard Bickerton, Gaia V Paolini, Jérémy Besnard, Sorel Muresan, and Andrew L Hopkins. Quantifying the chemical beauty of drugs. Nature chemistry, 4(2):90-98, 2012.
Rishi Bommasani. On the opportunities and risks of foundation models. arXiv preprint
arXiv:2108. 07258, 2021.
Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neurul information processing systems, 33:1877-1901, 2020.
Jinho Chang, Hyungjin Chung, and Jong Chul Ye. Contrastive cfg: Improving cfg in diffusion models by contrasting positive and negative concepts. arXiv preprint arXiv:2411.17077 2024.
Muthu Chidambaram, Khashayar Gatmiry, Sitan Chen, Holden Lee, and Jianfeng Lu. What does guidance do? fine-graines analysis in a simple setting. Advances in Neural Information Pro- cessing Systems, 37:84968 85005,, 2024.
Paul F Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. Advances in neural information processing sys- tems, 30, 2017.
Jacob Devlin, Ming- Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. Proceedings of the 2019 conference of the North American chapter of the association for computational linguistics: human language technologies, volume (long and short papers). pp. 4171-4186, 2019.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794, 2021.
httpp:://gi thub. http///c....ooa/aaaooooo//////// c-di ffusion conasioonccmooaiiiooooo coopossiiin on

---

Published as a conference paper at ICLR 2026
J. Dombi. A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems, 8(2):149- August 1982. ISSN 0165-0114 14. doi: 10.101600165-011449))90000000
Peiran Dong, Song Guo, Junxiao Wang, Bingjie Wang, Jiewei Zhang, and Ziming Liu. Towards test-time refusals via concept negation. In Proceedings of the 37th International Conference Neural Information Processing Systems, NIPS '23, Red Hook, NY, USA, 2023. Curran Associates Inc.
Randal Douc and Olivier Cappé. Comparison of resampling schemes for particle filtering. In ISPA 2005. Proceedings the 4th International Symposium Image and Signal Processing and Anal- ysis, 2005., pp. 64-69. Icce, 2005.
Yilun Du and Leslie Pack Kaelbling. Compositional generative modeling: A single model is not all you need. CoRR, abs/2402.01103, 2024. URL
103.
Yilun Du, Shuang Li, and Igor Mordatch. Compositional visual generation with energy based mod- cls. Advances in Neural Information Processing Systems, 33:6637-6647, 2020.
Yilun Du, Conor Durkan, Robin Strudel, Joshua B Tenenbaum, Sander Dieleman, Rob Fergus, Jascha DinhtDiccstein, Arnaud Doucet, and Will Sussman Grathwohl. Reduce, reuse, recycle: Compositional generation with energy-based diffusion models and memc. International con- ference machine learning, pp.
PMLR, 2023.
Jerome Eberhardt, Diogo Santos-Martins, Andreas Tillack, and Stefano Forli. Autodock vina 1.2. O: new docking methods, expanded force field, and python bindings. Journal of chemical information and modeling, 6188):3891338 3398, 2021.
Peter Ertl and Ansgar Schuffenhauer. Estimation of synthetic accessibility score of drug-like molecules based on molecular complexity and fragment contributions. Journal of cheminfor- matics, 1(1):8, 2009.
Timur Garipov, Sebastiaan De Peuter, Ge Yang. Vikas Garg, Samuel Kaski, and Tommi Jaakkola. Compositional Sculpting of Iterative Generative Processes. Advances in neural information pro- cessing systems, 36: 3::::2666 1666-12702 2023.
Sachit Gaudi, Gautam Sreekumar, and Vishnu Boddeti. Coind: Enabling logical compositions in diffusion models. In The Thirteenth International Conference on Learning Representations, 2025.
URL http://pwwwwweeweeee///////d//da
Antonio Ginart, Melody Guan, Gregory Valiant, and James Zou. Making ai forget you: Data deletion in machine learning. Advances in neural information processing systems, 32, 2019.
Jiaqi Guan, Wesley Wei Qian, Xingang Peng, Yufeng Su, Jian Peng, Jianzhu Ma. equiv- ariant diffusion for target-aware molecule generation and affinity prediction. arXiv preprint arXiv:2303.03543 2023.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Weinberger. On calibration of modern neural networks. In International conference on machine learning, pp. 1321-1330. PMLR, 2017.
Geoffrey E Hinton. Products of experts. In Proceedings of the Ninth International Conference Artificial Neural Networks, volume 1, pp. 1-6. 1ET, 1999.
Geoffrey E Hinton. Training products of experts by minimizing contrastive divergence. Neural
Computation, 14(8):1771-1800, 2002.
Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. In NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021. URL https://openreview.ndd forum/id=q@SAkxfYY.
Michael Jordan and Robert A Jacobs. Hierarchical mixtures experts and the EM algorithm. Neural Computation, 6(2):181-214 1994.
12

---

Published as a conference paper at ICLR 2026
Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361 2020.
Rafal Karczewski, Markus Heinonen, and Vikas Garg. Diffuusinn models as cartoonists: The curious case of high density regions. In The Thirteenth International Conference on Learning Represen- lations, 2025a. URL http:///weeeeeeveeeeeeeeoomnnnnd
Rafal Karczewski, Markus Heinonen, and Vikas Garg. Devil is in the details: Density guidance for detail-awaan generation with flow models. In Forty-second International Conference Machine Learning (ICML), 2025b. URL
Shervin Khalafi, Ignacio Hounie, Dongsheng, Ding, and Alejandro Ribeiro. Composition and alignment of diffusion models using constrained learning. In The Thirty-nimth Annual Confer- ence on Neural Information Processing Systems, 2025. URL https://ppeneevie..eee///uunat WO7rfbEGRU.
Michael Kirchhof, James Thortton, Louis Béthune, Pierre Ablin, Eugene Ndiaye, and Marco Cuturi. Shielded Diffusion: Generating Novel and Diverse Images using Sparse Repellency, May 2025.
Erich Peter Klement, Radko Mesiar, and Endre Pap. Triangular norms, volume 8. Springer Science & Business Media, 2013.
Yann LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/nmist// 1998.
Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, et al. aetrieval-augmennt gener- ation for knowledge-intensivy nlp tasks. Advances in neural information processing systems, 33: 9459-9474, 2020.
Nan Liu, Shuang Li, Yilun Du Antonio Torralba, and Joshua B. Tenenbaum. Compositional Visual Generation with Composable Diffusion Models. In Shai Avidan, Gabriel Brostow, Moustapha Cissé, Giovanni Maria Farinella, and Tal Hassner (eds.), Computer Vision ECCV 2022, pp- 423-439, Cham, 2022. Springer Nature Switzerland. ISBN 978-3-031-19990-1. doi: 10.1007/ 118-3-031-19790-11 26.
Christian A Naesseth, Fredrik Lindsten, Thomas B Schön, et al. Elements of sequential monte carlo. Foundations and Trends@ in Machine Learning, 12(3):307-392, 2019.
Thanh Tam Nguyen, Thanh Trung Huynh, Zhao Ren, Phi Le Nguyen, Alan Wee-Chung Liew, Hongzhi Yin. and Quoc Viet Hung Nguyen. A survey of machine unlearning. ACM Transac- tions on Intelligent Systems and Technology, 2022.
Tomohiro Nishiyama and Igal Sason. On relations between the relative entropy and chi-square- divergence, generalizations and applications. Entropy, 22(5):563, May 2020. ISSN 1099-4300. doi: 10.33906e22005633. URL
Bernt Øksendal. Stochastic differential equations. In Stochastic disferential equations: an introduc- tion with applications, pp. 38-50. Springer, 2003.
Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. Training language models to fol- low instructions with human feedback. Advances in neural information processing systems, 35: 27730-27744, 2022.
Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp- PmLR, 2021.
Marta Skreta, Tara Achound-Sadeehh Viktor Ohanesian, Roberto Bondesan, Alán Aspuru- Guzik, Arnadd Doucet, Rob Brekelmans, Alexander Tong, and Kirill Neklyudov. Feynman- Kac Correc- tors in Diffusion: Annealing, Guidance, and Product of Experts, June 2025a.
13

---

Published as a conference paper at ICLR 2026
Marta Skreta, Lazar Atanackovic, Avishek Joey Bose, Alexander Tong, and Kirill Neklyudov. The Superposition of Diffusion Models Using the Itô Density Estimator, February 2025b.
Yang Song, Jascha Sohl-Dickstein, Durk P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations (ICLR), 2021.
James Thornton, Louis Bethune, Ruixiang Zhang, Arwen Bradley, Preetum Nakkiran, and Shuangfei Zhai. Composition and Control with Distilled Energy Diffusion Models and Sequential Monte Carlo, February 2025.
Ramakrishna Vedantam, Ian Fischer, Jonathan Huang, and Kevin Murphy. Generative models of visually grounded imagination. International Conference on Learning Representations, 2018.
URL http:://pwwewweeeeee/////o/n//ca
Pablo Villalobos, Jaime Sevilla, Lennart Heim, Tamay Besiroglu, Marius Hobbhahn, and Anson Ho.
Will we run out of data? an analysis of the limits of scaling datasets in machine learning. arXiv preprint Ttt::2211.04225. 1:1, 2022.
Wenhao Wang, Yifan Sun, Zongxin Yang. Zhengdong Hu, Zhentao Tan, and Yang. Replication in visual diffusion models: A survey and outlook. arXiv preprint arXhv:2408.00001, 2024.
Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yo- gatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al. Emergent abilities of large language 2022.
models. arXiv preprint
Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li. Ming Ding, Jie Tang, and Yuxiao Dong. Imagereward: Learning and evaluating human preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36: 15903-15935, 2023.
Seniha Esen Yuksel, Joseph Wilson, and Paul D. Gader. Twenty Years of Mixture Experts. IEEE Transactions Neural Networks and Learning Systems, 23(8):1177-1199 August 2012. ISSN 2162-2388. doi: 20.1100//NN
Lvmin Zhang, Anyi Rao, and Maneesh Agrawala. Adding conditional control to text-to-image diffusion models. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 3833-3847,,0 2023.
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Section 3 Failure Modes in Score-Composition
Negation Instability. Inconsistent Temperature Scaling.
Section 3.1 Section 3.2
Violation De Morgan, Idempotency, Distributivity.
Section 3.3
Section 4 De Morgan Dual Dombi Composition Operators Properties Dombi Operators.
Section Section 4.2
Properties Stable Negations.
Section 5 Guarantees on Algebraic Precision and Stability Bounds on Idempotency and Distributivity Bias. Bounds on Dynamic Composition Stability.
Corollary 5.1 Proposition 5.2
Section Exact Sampling with Feynman-Kat Correction
Section 7 Experiments Image and Molecule Synthesis
Figure 6: Overview of the main contributions in this work.
A OVERVIEW OF THE PAPER
We provide graphic to summarize the structure of this paper and our main contributions in Figure 1, and restate this in more structured way in Figure 6.
B DE MORGAN DUAL DENSITY OPERATORS
In this section, we provide a general method for deriving a score calculus with the desired point- wise properties over the underlying (possibly unnormalized) probability densities. As described in Section 2.3, the semantics we seek align with "fuzzy intersection/coojunction" operators A, more formally known t-norms (Klement al., 2013). In order to obtain the "fuzzy union/disjunction" operator V corresponding to A, we choose to use De Morgan's 's law: x VVy = -(-x// A (-y) Here, - is a negation operator. Choosing De Morgan dual pair of operators allows for semanties- prretiivii-eeeeinggg rewrites (e.g., into negation normal form), which is convenient for implementing large compositions.
To connect theory fuzzy logic to our setting of density operators, we require lifting function, to map the target domain of densities, Ryo U foo), which we will denote with [0, oo], to the domain of memberships, [0, 1]. We use a lifting function á : [0, ooj + [0. 1] to accomplish this. As should maintain the semantics of high densities corresponding to high densities, we define it to be an order-isomorphism. Finally, we can use result (Dombi, 1982) define a class of De Morgan Dual operators on (unnormalized) probability densities.
Later in this section and in the main text, we implement these density operators through their induced score calculus, since diffusion samplers consume score fields rather than normalized densities.
Definition B.1 (De Morgan Dual Density Operators). Let à [o] [0, 1] be order- isomorphism and f [0, 1] > [0, oo] be a continuous, strictly decreasing f with f(0) = co. For o. we define for density function p
-p(x) o 6 ((p(x)))
(15)
Pi (x A (x = =9 (g(pp (x)) + g(p2(x)))
(16)
(x P2 (x == - px()))
(17)
By construction, we have the following properties.
Proposition B.2 (Algebraic properties (pointwise)). Assume &b [0, 0, is continuous order-isomorphism. and assume = o Φ [0, 5oj - [0, xo] is bijection with g( (c)) Define m, A, V as Definition B. I. Then for all E [0, oo]:
(10, oo), A, co) is commutative monoid.
((0, co), V. 0) is commutative monoid.
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is involutive: -(-u)
(De Morgan) -(u A 0) u - and -uu V -u
(Neutral Element) There exists unique u" (0, oc] with - w², given by u² (1/2).
B.1 DE MORGAN DUAL SCORE OPERATORS
As Diffusion models operate on scores, i.e., on the gradients of log-likelihoods, we need transfer our operators to the domain of scores. We use the composite generator function 9 f o & from Definition B.I. and sketch that De Morgan Dual Density Connectives A, V translate to weighted score combinations.
Lemma B.3 (De Morgan Dual Score Operators are linear combinations), Let & and f be fully differentiable and defined in Definition B.1. We again let g = fo Φ. For any two strictly positive density functions Pr, P2 with Pı A (g(p)) + 9(22)), it holds pointwise that
V Vlog an V lo + V log
(18)
where the scalar weights a; == w(p.)/w(p) A p2), with the function w(u) = (u). Importantly are dynamic "responsibilitty weights".
Proof. The statement follows from standard calculation
②(g) g(p)) + 9(22)
(19)
g' (q) Vq (p1) Vp1 ('22
(20)
(()) Vq (P1)
Vpı
Vp2
+ w(p2)
(21)
q
p1
P2
w (g) V log b = w(p)) V log w (p2) log
(22)
V log d V log + 022 V log
(23)
The statement for follows analogously, and for negations, we always rewrite for some density- dependent factor к V log( -p) = s(p) V log This, together with De Morgan's law, allows us to represent any nested formula over scores as a linear combination over (negated) base scores.
Lemma B.3 allows us to extend the properties in Proposition B.2 to score models.
Proposition B.4 (De Morgan Dual Score operators are linear combinations). Let F be any com- position function obtained by nested application of A, V, over density functions p , Pk. Then there exist scalar weights og such that
Vog F(p.....,
0 V log Pi
(24)
Proof. The result follows from the application of De Morgan's law to rewrite F into negated normal form, i.e., as an otherwise negation-free formula over (negated) literals. Then, og can be computed by repeated application of Lemma B.3, and positive and negative occurences collected together
(ps) and h(p;)) = --1(1 - o(p:)) as V log = k(p)) logp; la(ps)
with k(p;)=
B.2 REFERENCED NEGATION AND CONDITIONAL GENERATION
As we care specifically about the pointwise properties of our operators, can let the generators and & themselves vary. particular we may use @(p: x) as a location-dependent lifting function. This is particularly useful to draw parallels related literature in guided generation/avoidance. The choice of & defines our fixed point with respect to negation, and a semantically meaningful fixed point can be modelled with unconditional base model.
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In conditional generation, are interested in operations the conditional distributions (Du al., 2020; Liu et al., 2022). More formally, we are given conditional models p(x y)) for given concepts yı, well the unconditional model p(x) and are interested in sampling with formula F from
p(x F(yi..... yi ).. In the typical probabilistic interpretations, (Du & Kaelbling, 2024) implement these operations density ratios. This, in our setting would translate
(p(x 11) p(x)
p(x yk) p(x)
p(x P(yi....,. x p(x, F(ni...., - x(x)F
(25)
As the denominator p(x) is constant across the formula, we may absorb in into @ as constant nega- tion fixed-point. We then denote in the notation of the main paper with (p; x) == d [(x)
e(x] where
now p(x) takes the role of the conditional distribution, and c(x) is the constant negation fixed-point.
B.3 DERIVATION AND PROPERTIES OF DOMBI CALCULUS
In the main paper, we focus particular class of De Morgan dual density operators: the Dombi operators. We follow the definition of Dombi (1982), lifted to our setting, to define the Dombi
operators with (c(x) z+c ++1 and fx(x) ( 1)x and derive their corresponding score
calculus here First, we can see here that 0r (z) = ce = I g(x) = fx(0c(=)) = (4) I-x
Mxx)= = fx(1 (x)) s (++e) Sa
(अ) Further g --² cr -1/A With this
we can derive Definition 4. 1 as:
-p(x) 0- (1 dc(p(x))) Φ5 ( c(x)
((x) + p(x(() ) c(x)2
(26)
p(x)
(i(x) ^v p2 = y " (x)) + +((2((x)))
(27)
((ett))
[(xx) )a- (((((t))
c(xx)
1
-1/A
=x(x)
pp(xx)
(28)
1 d
-1/2
(mx()))
(29)
(x((x)
(x) + +xx(x) 1/A
(30)
(x()) VX P2(x) = -c( (x) ^x (x)2
(31)
x(x)²
(32)
c(x)²
^ a(x)²
P: (x] P=(x)
(33)
^x P=(
(X]
1
(34)
(p1(x)* A + (2())))-11A
= (xi(x)* (2())))) 1/a
(35)
In log-likelihoods and scores, the negation is straightforward. For a power-mixture (pi (x)* + (x() 1/a the log-likelihood and score operations are familiar. We investigate dis- junction and conjunction the same time and state for all X + 0:
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((x) - (pI(x)) + -22(x))) 1/2
(36)
1
log (xx)
log (pi(x)* + p2(x)²)
(37)
A
1
log (exp(Alogp1 (x)) + (xp()
(38)
X
(39)
A
Vx logg(x))
Σ
10g
(40)
(E(1,2)
(x( (1 ( P2(xx) Vx log logp
Σ (E(1,2)
(41)
In terms of score calculus, Dombi Operators, end up being softmax-weighter convex combi- nations of the component scores, in accordance with Lemma B.3. Further, they inherit the favorable properties in Proposition B.2.
B.4 DOMBI ERROR BOUNDS
For given value of A, the maximal difference between the Dombi operators and the min / max functions can be easily bounded as an additive term in ligeiiie::ooo::
Proposition B.5. Let Ax Vx the Dombi density operators. Then holds that
Vx,y E R20
min(x,1 y)2 1/A < x Ax S min y} max(x,y)
(42)
Yx,y E Rz0
V y max 2/A
(43)
Proof. See Appendix C
C PROOFS
Proposition B.5, Let Ax, Vx be the Dombi density operators. Then it holds that
Vx,y E R20
min y)2-1/A S x Ax y S min {z. y} max(x, y) < x V < max y/ 1/A
(42)
Vx,y E R
(43)
Proof. We show the case for p Vx q q= (p> +/*) 1/a first. The definition of V is equivalent that of a P-norm over two components. We have the standard inequality (w.loo.g. for 2 ②)
V> V> = (p> + < (2p*) 1/2 2/2 max o
(44)
The lower bound similarly follows from
VV = (p> + /2)
1/A 2 ( max {p. 4)
(45)
For Ax, we can use De Morgan to obtain the symmetric bounds. We can note that the upper bound tight for q and the lower bound is tight for q = 0.
O
Proposition 5.2 (Mixture Stability). Let softmax, (Alogp' (Xllga p² ox p². Then holds for the score difference As st s
Alog p"), p²). dombi composition
E|da;
xx]1
0(I22 - >/o (
d(a), = O(A O(
Proof. Our objective is to bound the rate of change of the stochastic mixing weight a. We use f = log pf log Ps and as" (1 a) 2 As s 52 shorthands for this proof.
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First, compute de and then use this to bound the rate of change for sigmoid(Ae). By
Equation (3), we have for u = fft + o²s
dl (dr, As) (As, Jt) di oi
(11=111--11) 11x2211) dl
(46)
of
= (As, u) dl + 0z A8, dw) + (As Si) di 2 -1=211)) di
(47)
0. (1|{ || dt 3e(As,dw)
(As,s))) dt
(48)
2
of (As, 2s (s] + 3))) dt + or(As, dw).
(49)
2
And, for later convenience, we explicitly state that d[f] = of As
We now apply Ito's Lemma d'((ee) = (l,) dl, + 20 (4) d[e]: ((t) sigmoid(Ae) and get
da = ===a - --(1 de + 2
aaa(1 - a) - 20) d(f).
(50)
Finally, we combine terms
da = Aa(11 -a) o 0 (As, 2s - (s) + s;)) dt + oAs
1 + 2 A²a(1 (06(1-)(11-- - 2a) As| dt
(51)
Aax(1 a) dw) + oi
a a)) (2(4s, 28 - (5: + o;)) + 22 20)1 4s12) dt
(52)
2
of
= =Aa(1 - a) dw) + a(1 a1 2a² - A) || |||ه dt
(53)
2
where maxae a1 2a)
v3 18
Hence, the drift satisfies
v/.3 |Xa \||| As|?
R[da, xx|] S
36
and the quadratic variation satisfies
X²O7
dla). = X2 -aa==> di S
| 11 dt.
16
C.I FEYNMAN-KAN KAC CORRECTION
The reweighting equation
dw; g(x) 0p.(x)
(x)
(54)
describes how the log- neg-weight-fin field influences the marginals of the weighted SDE. The translation of continuity (drift) terms and diffusion terms into log- log-weights is then given by the following schemes:
@p:(x)
(V, Pe (x)),((x))
(p(()) (V, (((x))((x))) p((x) =
(55)
dw, (- (V, (x)) (Vlogpe(x), (x()))
for drift terms and
(x)
0² Apr(x) 2 p((()) (A (x) + |/ logp;(x))13) =
a²
Øt
(56)
dw
(ologp: + (olg
for diffusion terms.
Dombi Composition is equivalent to applying a power-norm to probability distributions. We recast this annealing, case shown by Skreta et al. (2025a), then taking an (unweighted) mixture and then inverse annealing of the mixture of annealed distributions. We state the following results before proceeding with the main proofs.
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Lemma C.1 (Mixture of SDEs FKC). Consider two weighted diffusion models qi (xx, (x) de- fined via the Feynman-Kee Kac equation with corresponding weights g. (x), (5).99(( (x). The weighted SDE corresponding to the sum of the marginals px(x) x q: (x) + (x), with of oF (x)++0(1 (0,1)
o((x))
dx:- [-f((x)) + of Vo a. (x,) + a² V log of (xx)) dt + ordW:
(57)
dw,- = (x) + 9 dt
Proof.
We have, for (x)
at o
Op:
0qk
0uf
0q:
duF dx
(58)
01
0
00
01
= a (x)( ft v log क) (x)) + o3
Agi (x) + (x) [x)] +
2
(59)
o²
Agi i (x) + (x)) [x] - -f
dyt
ag
(V.9((x) 1 of v log (x))) +
+
dx
2
di
V2 (x))) + o
=
=(V, o. /e (x)
A (x) + (x) |o (x)] +
2
(60)
V4 (x))) + oi
(x) + (x) [x)] - -1
0gl
0q dx
(v.o)(x) oo
of (x)
01 de
(V.9 0. (x)
(x) + (x)( 1-fe - oo (x) (xi())))
1
(61)
o7 (x) + (x) /
opc dx
2
1 (f
= =(V, (ot (x) (x))( fi) + at (x) (-o? [x
+
(x)
(x))))
(62)
of
Axx (x) - / dpc dx
at = (V. (g) + (x(())))- f1) + (-o,
2
+ (x))))
of Ax:(x) +xx(x) - ff
Opp dx at
(63)
2
(V. (gl (xx + (x)( fo) px(x)) (To2
[x) (x(x))
+ [x]
(x)) )
(64)
of (p:(x) + (x) - -f Üpt dx
2
Øt
= -V (x)( J0 (x)) (-o2 (x)
+ V (x))
[xx(xx) (x(x)
))+ )
(x(x)
(65)
of (x) + +xx(x) - /
Üpe dx
2
at
= <xV
(-fe fe of
Vet (x)
vgR(x)) )) + o2 px(x)
Apx(x) + (x)))))( - f
ff
dpp dx
(66)
(x(x)
2
it
(-fi-ot - of (x)(x)
v log (x) + (x(x) v log oी (xx) +
= (V)
(x()))
px(x)
(67)
of (x) + +x)(x))((x) - f
Öpc dx
2
at
(x) (-fi ft- - of (at V log qt (x) + of V log o
(x) + -x))xx)( - /
Öpp dx
(68)
2
at
of
(x) (-fi of of (at loggl( of (og
Ap(()) + (x)x)
(69)
2
We can simulate this as
dx,- [-f((x;) + 02
+z) (x))) di + a.dw.
(70)
dw,- = [a] + (x)] dt
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Lemma C.2 (Target Score Annealed SDE + FKC, Skreta et al. 2025a). Consider a diffusion model qe (x) defined via the Feynman- Mac equation with the weight-field ge(x) and some parameter X R {0}. The weighted SDE corresponding the annealed marginals pe(x) (x(x)* can be
performed by simulating the following weighted SDE
dx. = [- f((()) + Vo (xx)) dt + o,dW;
(A - 1)(V. + A(A 1) -h | log (x)] + )g(x) dt
(71)
dw
Proof.
We follow the proofs of Skreta et al. (2025a).
We aim find the partial derivative of the density Pc(x) dt over time @p.(x)
9x(x)*
8t where
of 0q. (x) - (V, (V,:(x)(( + Voglo Axx(x) (ox Øt
Then we have
0 log (x)
dx((x)
(72)
Øt
(x))
Ot
(V, +f (x)) + o: Aq((x) 4(()
+ g(x)
(73)
(x)))
2
(V, (x(x)((()))))))))))))))) of (x))
A log + V gogge(1))) +(x)
(x(x)
(74)
-V, -f + o, v logg)) (-fe of V loggt, Vlog ④))
oi
(A log 4: Voo q: (12) 5xx)
(75)
(V. + (Jr, logg:) - 0/ A log q: - of V 1
o
+ 2
A log q: + V log qt 12 + g(x)
(76)
o/ ( A log + || log 1= g(x)
(v) + (Je, V log
(77)
2
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and can now compute
log
a llg (x)
- / Xx)(xx)
aoog (xx)
dx
(78)
Øt
0t
= = fi) + (fe, Vloggi) ge)
o²
+ o ( | (.12)l1))) + 0
(79)
/ (x) alog (x) dx
do? (A log qi + log V + Ag-
= Ab) + NLee, Vlog g))
(80)
/ xxxxx) (x) dx
Ot
Ao?
AV)) (Fe, Vlogp))
(A A Alog qu
|/ VVlog (1) Ag
2
(81)
/ (xp((x) alog (x)) dx
= fV) + (fer Vlogpt) - (1 - ft)
log q: + Volg + AØ-
f Pa V. + - (1 fa)
oo?
A log qi |/ log (13) Ag dx
2
(82)
[VV + (fo, V Vlggp)) - (1 - X)(V, fc)
A log Ø: + IV log + 2o
2
(83)
-f p - - \),, oo, (A log + log q: 12) + Ag dx
o²
ai | V log + Ag
-V) + Vlog.)) -(1 A1),
A log Pt
2
2x
(84)
-f pb (1 A) 03 A Alggp
07
V log pi 11
+ Ag
dx
2X
ft) + (f,, Vlogp:) -11 - {)(V,
A log Pt
|
log pel
2
2
(1
2 2
1 of
1og +Ag - -f pd
(1 - () fi)
of
log pt
o?
| log pt 2
2
(1- X
1 of
V log + +10]dd dx
2
(85)
o7 A log
o7 V
ft) + (Se, Vogg - (1 - A.)
log 2
2
2
(1 1 ot VV +\g -f po -1- A)(V,
(86)
1 1 1) 2 Voogpe + +29]dx dx.
of
With this, defining g' = -01-1)( (4) + +(1 - 3) V 1 + >g we finally have
i log Pt
oi A log Pt
02
4 - px(x)g' -/mxx))g' (87)
=(V, fi) + (fr, V Vlogp))
V
log 12 +
0
2
2
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0p: Pt
i logg Pr.
(88)
0t
Öt
oi
of
= (V) + (Se, V logp:)
A log
V Vlggplll + gx()) - Egggg (x)
(89)
2
of
of
= - -fpp +
A
log
V log pell2
- g (xxx) - Ep. [xxx
(90)
2
o7 Apt.
oi
of
- (V. -.ppa) +
+
V
Po 1
v log + g'(x) - Epg
(x)
2
Pt
2
(91)
oi +
-V of V logg:))
Apr + [g' En,g (x)]
(92)
2
And finally, can reexpress this as
Öp. Ot
of Ap. + Pe (g((x) - Ep.d (x))
= (v
+ 2
(93)
And for A > 0 we can simulate this as
dx; = [-))(x) + (o di + a,dw;
- (VV..((x) + A(A 1) o7 / log 9. l0 2 +2g dt
(94)
dw, = d((x)dt=
Proposition 6.1 (Referenced Negation CFG+FKC, Skreta et al., 2025a). Consider two diffusion models of (x), (x) defined via the Feynman-Kan equation Equation (6) with weights gi The weighted SDE corresponding to the referenced negation of Pe(x) & (p (x) is, with du, = 9x))dt
dx; = [xf((x;) a2 log (xx) - log qf (x:)) dt + a.dw;
(13)
(x) = V log / (x:) V log 9f (xx) 29) (x).
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Proof. We start with the annealed distribution oi xx)) and the annealed pseudo-distributin %/ (x) xx)
We now try find
a logg: d log qi
log gt
@ log of
log af
dt
dt
pt
2
a
d a llg
(95)
a log q
a ol
i log oج
(96)
d1
dt
0
dt
=2
1 f() + (ft, V log )
A log qi + |I of (=1)) +x(())
2
σi
[V) (f., log
A log qi
V 4f (1² + (x)
(97)
/ p((x))
3 log (x)
b Blog uf (x)
dx
Öt
Öt
of
(V. VV, d
(Se, V log 9 + 2
(A (Alog log qi + || log 1oggol17 121) + +9 (x)
lo + VVlo + g (x) of
- px(xx) [2 -/m(x0)
100lo18(x) (x)
I log
2 Blog qi (x)
dx
öt
at
(98)
o²
= fV) + (fe, Vlogpt)
(2(A log of + V log log (13) ) - logg + ||
2
(99)
θ log 1o (2 8 Ölog at (x) Öt
2x) xx(x)- - p((x))
2
dx
ôt
of
S1V + (/. Pz)
llog Pe + | pel? - 2/ V log Vlogg/12)
2
(100)
2g²(x) -'(xx- - / p((x)
1 log loggi(x) (x()
Ölog al (x)
2
dx
Ot
Ot
ft) + (fe, Vlogp4
(A log + | VVlog Pegl pell ) + of Vog Vlog (101)
2
2g¹ (x) - nو (x) Ep. !o? | log
V Vlogal + (x) s'(x)
And with g(x) = V| V log of V Vlogo! 12 2x) - gx(x)
Op: = Pt
OlogPt
(102)
Øt
Øt
oi pi Jt) + (fi, logpi))
A logg: |1 [V1ggn1113 + pi [0(x) [x)
(103)
2
oi Apt + Pt [o(x) Px.((x)..
(x V log
(104)
which we can simulate with
dx; (x, (2V log क2 (x,) - V log ( dt o.dw.
(105)
(x) = V log पt (x:) V log 9² (x:)11 + 29 (x) (x).
Theorem 6.2. Consider weighted diffusion models of (x), (x) defined via the Feynman- Kac
equation with weights gf (x), ot (x), and a parameter X E R [0]. The weighted SDE corresponding
to (x( x (x)) + of (x))) 1/2 with of x[())* (x)x
4/ (x)
E (0, 1), and dw; = (x))dt is
dx, (x)( + of (at V log of (xx) + ag v log oi (xx))] dt + ardw.
(((x) (1 2) 0²
Σ (E(112)
aj log
Σ
af V log (o (xx))1
+ Σ
(x)(
(E(1.2
(14)
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Proof of Theorem 6.2. We now our two lemmas to show the main result. We begin with
dx; [- (-x((x)) log o (xx)] dt o,dw,
dmp (A-1) (V. + d²
(106)
2 ll dt + >g((x)
for both annealed distributions, according Lemma C.2. Then, by Lemma C.I, we have mixture of these distributions with
dxg [- fxx) + of log / (x.) a. log of (xx))] dt o,dw
dw,- =a! (A 1--1 1) (V.)((x))) + 0²
l/ (x)) (x:) 112 dt + (x)
2
(107)
at (A (x-1) 1) (IV.A((x))) +
+²
8²
+ {x]
which simplifies to dwy- (A - 1)( +
Σ og (a-11)
0² _ 2
V log of (xt)lldtt + o(cxe))
(108)
(1,22)
Finally, we apply Lemma C.2 to the resulting mixture with 1/A. This then results in dx, -- f((x) + of (a. V log af (xx) + a log & (xx) dt + ordWe, which is the target score as desired. For weight-field we then have
(109)
( -1) (VV.(((()))))+ + 0² oo de+
dwi
1)(V f(x()) +
Σ at (a-
0² _
[Vlog (xx)) dt
(E(1.2)
(110)
1- f(xx))) + 1- oo² | log (xx) Muso log (x:))l2dt+
A 2 X
(a- 1) a²
A (V.J((x)))dl +
E ai
i llog (x:))( gi di
(E(1.2)
(111)
(1- A) bo2
0²
Σ |E(1.2)
ai log (x)) dl+
2
(1 1) 0²
Σ
V of (x1) g((xx) dt
of
(E(1.2)
(112)
0² = (1= 2
a v log
Σ (E(12)
afll
dt Σ
(xg)
(E(1.2)
(E(L2)
(113)
We can see that, as expected, for À = 1 we are left with the unweighted mixture of distributions. For more complex compositions, the weight fields just propagate well, that the statement trivially generalizses to more than two diffusion models, so we maintain associativity.
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(a) (p1 VP2) (-p1 V -p2) Aps
(b) (PL V-D2)A
(c) (p2 Vps) ( (-P2 -ps) Ap
(d) (-pa--s)
(c) ps PPI -22
(f) pi A p2 Ap
Figure 7: Generated MNIST score compositions.
D EXPERIMENTS
All our experiments on stable diffusion and SBDD were performed on unmodified, pretrained mod- els. We performed inference Nvidia v100 and a a100 GPUs.
D.1 MNIST EXPERIMENTS
We reproduce the setup of (Garipov et al., 2023), and generate images from the score composition of the three toy mnist models. The code to training the models can be obtained from the code repository and training was performed on Nvidia GTX 3080 desktop within 10 minutes.
We show image collages for non- non-trivia trivial example formulas in Figure 7. For each formula we gener- ated batch of 1024 images.
D.2 STABLE DIFFUSION IMAGE GENERATION
We reproduce the stable diffusion experimental setup of (Skreta et al., 2025b) with Stable Diffusion v1-4 available pretrained publicly at huggingface: https: https://huggingfacee co/ hwtomiisssstaaledifffiieee PoE, superdiffss and as well as joint prompts.
We then report
We use 20 pairs of conjunctive prompt pairs and generate 20 images each. We provide a batch of the generated images in the supplementary material and list the prompts here, also reused from (Skreta et al., 2025b):
a mountain landscape" A Msilhouette of dog"
a flamingo" A "a candy cane"
"a dragonfly" A "a helicopter"
"dandel lin"" A "fireworks"
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"a sunflowez" A "a lemon"
"a rocket" A "a cactus"
"moon" A "cookie"
a snail" A "a cinnamon roll"
an eagle" A "an airplane"
zebra" "barcode"
chess pawn" A "bottle cap"
a pineapple" A "a beehive"
"a spider web" A "a bicycle wheel"
"a waffle cone" A "a volcano"
"a cat" "a dog"
"a chair" "an avocado"
"a donut A "a map"
"otter" A "duck"
pebbles on a beach" A "a turtle"
"teddy bear" A "panda"
For the contrastive Prompts, we partially use our own prompts and partially use the prompts from (Dong al., 2023). We provide a batch of the generated images in the supplementary material. and list the prompts here:
"A night sky with stars and a crescent moon, reminiscent of Van Gogh's 'ttarry Night', A- "Van Gogh"
"A night sky with stars and crescent moon, reminiscent of Van Gogh's 'Starry Night'. A- "Picasso's Cubist style"
"A portrait of a man with a distorted and fragmented face painted in Picasso's Cubist style." A= "Picasso's a Cubist style"
"A cat and ball on the shelf" An "cat, ball"
"There are bicycle and car in front of the house" A- bicycle and a car"
orange fruit" A- "orange color palette"
"a banana" A- "yellow color palette"
an ocean" A- "blue color palette"
"strawberry" A- "red color palette"
"round shape" A- "circle"
D.2.1 ADDITIONAL RESULTS
We provide additional plots illustrating the behaviour of composition under varying values of À in Figure 8.
D.3 ADDITIONAL RESULTS ON SBDD MOLECULE GENERATION
We report sweep across three values of À for the molecule generation task in Table 3. As the variance in this experiment is high, none of the differences can considered significant.
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Log-Lite Log-Litellhod Bias
Modure Var Warigiiity
0.3
0.0
250
1000
250
500
750
1000
Generation Step
Generation Time Step
SupDit.
PoE.
(u.9 ours.[1]
oure,110]
Mathod
SupDit.
PoE.
ours
(a) Variability of mixture coefficients for conjunction (b) Absolute difference in likelihood during generation
Figure 8: Mixture Stability vs Likelihood Bias in SD experiment. Figure 8a shows the absolute changerate of a², Figure 8b shows the median absolute log-density ratio. PoE (or geometric mean) has constant mixture coefficients, but log likelihoods diverge during the diffusion process. Superdiff forces equal likelihoods as the cost of highly variable mixture, especially early during the diffusion process. Dombi composition (ours.[A) provides a tradeoff, depending on X.
Table 3: Docking Scores of generated ligands for 14 protein target pairs (P1, P2), in batches of 32 ligands for 5 molecule lengths each. Extended runs across temperatures 7 (1, We compare conjunction with Dombi with various À with and without FKC with annealed base distribution and also report TafgetDiff from (Guan et al., 2023) baseline.
Method
Temp
PKC7
(2)
maxiP (.)
Better than ref.
Div.
Val. & Uniq.
QED
SA
TargetDiff
02.19
0.32
0.96
0.57
0.59
Dombi
0.3
68.
-7.57 2:311
0.26
1.37
0.88
1.300
0.96 1iin
0.58
2m
0.59
2.10
0.28
0.88
0.58
Dombi
67:92
7.53
2330
0.28
1.031
0.88
1:301
0.96
0.57 2.15
0.59
0.10
72.00
7.51
0J87
0.95
0.56
Dombi
72.83
7.71 1:105
0.27 1.035
1:303 0.86
0.95 not
0.57 6115
0.59
601
70:01
7.50
0.28
0.58
0.61
DomN
0.3
72.54 20:1
7.07
241
0.32 2:035
0.88
2:202
0.53 0.15
0.50
0.15
0.61
0.00
71.36
7.39
DomN
78.75
72.02
7.74
2445
0.37
0.31 1:0.5
0.88
2:202
0.54
0.15
0.60
0.59
0.62
009
DomNi
81.63
25.91
8.25
135
0.38 1:040
0.85
2:211
0.53
0.077
0.59
2.122
0.62
0.00
Dombi
-8.40
0.40
0.85
0.94
0.57
0.62
28

---

Published as a conference paper at ICLR 2026
E EXPERIMENT FOR COMBINATORIAL CONSTRAINTS
E.
SETUP
We illustrate the capability of Dombi compositions to adhere to combinatorial constraints by sam- pling uniformly from satisfying variable assignments of propositional formulas. For a formula with k propositional variables P., for E [1, ke), set up our diffusion ensemble follows: In IR place 2 Gaussian modes, one for each possible variable assignment. Then, in our ensemble, each of k score models simulates one propositional variable. For i E [1, we have access to 8ū, which defines a denoising process to uniform mixture of the 2 Gaussian modes, where the P: is true. Additionally, a reference model defines a denoising process uniformly all 2 Gaussian modes. For = 2, this setup is visualized in Figure 9a.
P1
XIR
Tall
Tall
PPL
Tlomm
Tiall
PFL
Domhi
1,00
0.58
Maja
Prod
1.00
0.00
0.00
Intial
hital
XOR
Prod
1.00
0.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
Dombi
Dombi
Product
Product
Oneloot
Dombi
1,00
1.00
0.09
0.99
Pood
1.00
0.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
(a) SAT experiment in R².
(b) Overview of SAT experiment for three formulas.
Figure 9: Figure 9a shows the SAT experiment in R2,, with squares corresponding to satisfying assignments. The corresponding numerical overview for k {2, 5, 103 in Figure 9b. Best are bold.
Our objective is then use score-composition to uniformly sample from satisfying variable as- signments. We repeat this setup for the Dombi operators, well as PoE/MoE composition for three formulas for E [1, 10), and report mode coverage, uniformity, and stability of the composition.
E.1.1 SAT FORMULAS
We three different propositional formulas: majority, xor, and one-hot. The formulations of these formulas are designed to test different aspects the score composition.
Majority
We define the formula over l variables as
Maja (P P&)
V
SC [S1-14/2]
..)
P.)
PES
This formula is negation-fiee free, but might lead to mode dropping for variable assignments with fewer positive variables.
One-Hot We define a formula where exactly one variable has to be true as
OneHot, (PI. Pt) V P.) ^ ( (-P.V-P,))
ISKjjR
is only quadratic in the length of the variables, but it contains many clauses without positive literals, requiring precise handling of explicit negation.
Exclusive Or
We define parity function over lt variables
(P. Pk)
A (E (mod
(o.7-P3 P.).
1=1
>,
This formula can only be expressed exponential length with 2 clauses, which explicitly ex- clude assignment with even parity.
29

---

Published as a conference paper at ICLR 2026
E.2 SCORE MODEL SETUP
We translate each of the 2 propositional variable assignments to Gaussian mode in R as
p(x) 2
Σ
Ne(xx/u 2,
RE(0,1}
We then define "directional" diffusion models
vi E [1 px(x)
Σ eEf0,11 0,=1
(x(x|4u-22 10144-2,,00 2,0²)
2
In this setup, each distribution plays the role of one propositional variable. The distributions p; can then be composed mirror propositional formula, with the goal that particles converge only to modes that correspond to satisfying variable assignments. We use as an additional stabilizing model guide particles any location that corresponds to an assignment.
As these models are mixtures of Gaussians, we derive optimal scores and energy functions from the standard Gaussian to our distributions in closed form.
We then model each type of formula for k E [1, 10] as direct composition and simulate 2 particles over 100 denoising steps.
For each mode, when then check Too hounding box around its mean of sidelength 30 and consider all particles within that radius to valid assignments.
In Figure 9b we show the most important metries: Neom the fraction of particles within bounding boxes of satisfying modes, Tall the fraction of particles converging to any mode. Additionally, we measure the normalized perplexity in the particle distributions across as PPL. In this experiment, PPL measures mode uniformity, where a higher number indicates more uniform samples from sat- isfying modes of the formula. In formula with K satisfying variable assignments, for batch of particles, with Mioorr particles within satisfying modes, we denote the fraction of partieles within the bounding box of the assignment index [1, K] 7) with E, f): Jur.. We then calculate PPI. for mode confusion as
PPL. 2(- Σx 2K
loga veen //K.
E.3 RESULTS
Figure 9a shows samples of formulas in R An overview of the experimental results is provided in Figure 9b. We can see multiple shortcomings of products in our experimental results. On the negation-free Maja. PoE. drastically reduces the por-mode variance, as seen in Figure 9a, drops most of the modes for Operators do not drop modes and maintain mensions. For XOR, and OneHot, PoE breaks down for & = 2 already, due to the negated literals. In Figure 9a, the modes of the PoE sample appear drastically biased by the negated clause. Some- what surprisingly, the Dombi composition can sample comparatively well from the exponentially sized XOR io. and struggles much more for OneHot, which is comprised of many purely negative clauses.
5, and completely breaks down for k: 10. contrast to this, the dombi
tose-too- distribution over modes in high di-
30


Reviews:
[{"review_id": "tZgmxAAfop", "reviewer": "Reviewer_9bJN", "summary": "Combining or negating (conditional) diffusion models is an important problem for controllable generation, whether in text-to-image generation or designing molecules which (do not) have certain properties.   The paper highlights pitfalls with several existing heuristics for combining models via the product or negating models with quotients, and seeks to correct these by turning to fuzzy logic operations.    The proposed Dombi Operators satisfy DeMorgan's Laws and encompass several existing heuristics for combination or negation.     Finally, the authors derive a practical scheme for sampling from the combination of diffusion, combining techniques from several recent works.\n\nThe authors demonstrate gains from the proposed compositions on text-to-image generation and dual target drug design, along with several insightful examples to elucidate properties in Figure 2.", "questions": "I am confused by the equation in Line 204, where the first and third logical statements are equivalent but the second is not.   I presume the authors wanted to obtain different \"probability-distribution logic\" for equivalent logical statements.\n\n\n\n*Minor comments:*\n- Prop. 6.1 should reference the Feynman-Kac / weighted PDE in Eq. 6.  Otherwise it is not clear where $g_t^{1,2}$ are coming from.\n- why introduce the notion of energy?  I thought it might be cleaner to just use densities.   \n- In Alg. 1 line 4, the authors point out the transformation $\\lambda \\leftrightarrow -\\lambda$ between conjunction and disjunction for fixed $\\lambda$.   This is a useful observation worth emphasizing!  \n    - I suppose it's obvious from the score expressions in Lines 8-9, but less obvious from the distributions\n\n**Observations:**\n\nA distinct justification for the Dombi conjunction and disjunction in Eq 8-9 can be given using $\\lambda$ (a.k.a. '$\\alpha$' or power-) mixtures or quasi-arithmetic means of densities ([1], used e.g. for annealing paths in [2]).      For a similar generator $h(p(x)) = \\frac{1}{\\lambda}p(x)^{\\lambda} - \\frac{1}{\\lambda}$, define\n$$\n\\mu^{(\\lambda)}_w(x; p,q) = h^{-1}[(1-w) h(p(x)) + w h(q(x))] / Z \\\\\n= [ (1-w) p(x)^{\\lambda} + w ~q(x)^{\\lambda}]^{\\frac{1}{\\lambda}} / Z\n$$\nNow, for $w=1/2$, this term factors out and scales the mean by $(\\frac{1}{2})^{\\frac{1}{\\lambda}}$ (similar to the factors in Corollary 5.1).   This $\\lambda$ representation is closely related to the $\\alpha$-divergence [1, Sec 4-5] (c.f. $\\chi^2$-divergence in Eq. 10, see below)\nThis may be frivolous, or may be useful for the following reasons:\n\n- $\\lim_{\\lambda \\rightarrow 0} h(p(x)) = \\log p(x)$, so $\\mu^{(\\lambda=0)}_w = p(x)^{1-w} q(x)^w / Z$, matching geometric averages or classifer-free guidance\n    - for $w=1/2$, we get an `idempotent version' of the product ($\\wedge$), i.e. $\\mu^{(\\lambda=0)}_{\\frac{1}{2}} = p(x)^{\\frac{1}{2}} q(x)^{\\frac{1}{2}}/Z$\n- Continuing with $w=\\frac{1}{2}$ and emphasizing the $\\lambda \\rightarrow -\\lambda$ relationship as in authors' Alg 1 Line 4, the product now fits nicely into the rest of the special cases described by the authors:\n    - max:  $\\lambda \\rightarrow \\infty$\n    - mixture (MoE disjunction)  $\\lambda = 1$\n    - Dombi disjunction:  $\\lambda > 0$\n    - product (PoE conjunction) $\\lambda \\rightarrow 0$\n    - Dombi conjunction:  $\\lambda < 0$\n    - harmonic mean:  $\\lambda = -1$\n    - min:  $\\lambda \\rightarrow -\\infty$\n- Perhaps it is not obvious to introduce $w \\neq \\frac{1}{2}$ from the fuzzy logic perspective (?), but quasi-arithmetic mixtures could expand the search space for effective model combinations.    \n- Logic is probably suitable for distributivity and chaining operations though.\n\n\n*Garipov et. al 2023 Contrast?*\n\nGiven my concerns about negation as presented (and for fun), consider a different way to derive Garipov et. al 2023's contrast operator.   We want $p(x)$ 'and' $p(x)^2/q(x)$ (high p, low q, needs to be integrable)\n\n$$\np(x) \\wedge_{\\lambda} \\frac{p(x)^2}{q(x)}  = \\frac{p(x)^3 / q(x)}{[ p(x)^{\\lambda} + p(x)^{\\lambda} \\frac{p(x)^{\\lambda}}{q(x)^{\\lambda}}]^{\\frac{1}{\\lambda}}} = \\frac{p(x)^3 / q(x)}{[  \\frac{p(x)^{\\lambda}}{q(x)^{\\lambda}}\\left( q(x)^{\\lambda} + p(x)^{\\lambda} \\right)]^{\\frac{1}{\\lambda}}} =  \\frac{p(x)^2}{\\left( q(x)^{\\lambda} + p(x)^{\\lambda} \\right)^{\\frac{1}{\\lambda}}}\n$$\nwhich matches for $\\lambda = 1$.\n\n\n\n\n\n[1] Amari 2007, Integration of Stochastic Models by Minimizing α-Divergence\n\n[2] Masrani et. al 2021, q-paths: Generalizing Geometric Average Paths using Power Means", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 4, "strengths": "The paper is logically structured and well-executed in (i) demonstrating the pitfalls of existing heuristic for model combination, (ii) deriving Dombi Operators which satisfy DeMorgan's Laws as we would expect from logical operations (iii) deriving an SMC resampling scheme combining two recent methods (Ito Density Estimators and Feynman Kac Correctors), and (iv) demonstrating its efficacy.\n\n\nSec. 3.2 makes a *very* nice point about implicit temperature scaling of score addition!\n\nI am very positive on the paper apart from a technical concern about the negation derivations (described below).   If valid, this should be mathematically salvageable but also may change experiment results.    Thus, I am giving a low initial score, but I am looking forward to discussion with the authors.", "weaknesses": "**Concerns re: Negation** \nI have a concern regarding negation in Definition 4.1, especially moving from constant $c \\perp x$ to $c(x)$.  \n- First, even for $c\\perp x$, I get $\\phi_c(x) = \\frac{x}{x+c^2} \\implies \\phi_c^{-1}(y) = c^2 \\frac{y}{1-y}$. \n     - Now, $\\phi_c^{-1}(1-\\phi_c(p(z))) = \\phi_c^{-1}( \\frac{c^2}{p(z)+c^2}) = c^2\\left( \\frac{c^2}{p(z)+c^2} \\right)\\left( \\frac{p(z)+c^2}{p(z)} \\right) =\\frac{ c^4 }{ p(z)}$\n     - This does not match the stated $\\neg_c p(z) = \\frac{c^2}{p(z)}$ in Eq. 7\n\n- What would it mean to take $\\phi_{c(x)}(p(x))$?   This is using a different function $\\phi_{\\mathfrak{c}}(t)$ depending on the input $t \\in [0, \\infty]$ (since for given $x$, we are interested in $\\mathfrak{c}=c(x)$ and $t = p(x)$ )\n     - It is also unclear how to invert $\\phi$ now\n     - Not sure where the `order isomorphism' condition is used in Def A1, but again, now $\\phi$ isn't a fixed function.\n\n*Nevertheless, the conjunction and disjunction distribution operations in Eq 8-9 are correct for constant $c$.   Thus, only the negation operation is problematic*.   If the main goal for this definition of negation was to derive the contrast operator of Garipov et. al 2023, there should be other ways (see Observations below).\n\n**Clarity**  \nThe clarity of the paper is further lacking in several places.\n\n*Definitions* (mostly fixed by including Def. A1 in main text)\n- L215:  The role of $\\phi_c$ has not been introduced.\n    - $N(x) = 1-x$ corresponds to $\\phi(x) = x$, and it probably does make sense to start with this $N(x)$.  You can then mention more general $N_\\phi(x) = \\phi^{-1}[ 1- \\phi(x)]$ and reference App. Def A1  (*not referenced anywhere in main text!*)\n    - This would also emphasize the role of $\\phi$ *alone* to define negation, which then feeds through to conjunction and disjunction via DeMorgan's laws for a given $f$\n- App Def. A1:  should have citation or proof that these satisfy DeMorgan's Laws (I confirmed that both hold for the given definition)\n- L242:  $g(x)$ is not defined.  \n\n\n*Sec 5:  Bounds on Precision and Stability*\n- Lines 274-275 and L276-277 should be reworded.   Only after reading the title of Corollary 5.1 \"Idempotency and Distributivity Bias\" did I understand what we we talking about.\n\n\n*Algorithm*\n- In Sec 6 (and possible Alg 1), it would be useful to re-emphasize the use of the Ito density estimator (Eq. 3) to track weights in the composite scores.   \n\n*Experiments*\n- What $\\lambda$ are used for operations in Figure 3?\n- Please check the logic in Lines 396-398 and consider writing the result of each logical clause for legibility.  I get that $p_{\\text{xor}} = \\{ \\textcolor{blue}{2,3}, \\textcolor{orange}{0,1} \\}$ so that $p_{\\text{xor}} \\wedge \\neg p_3 = \\{ \\textcolor{blue}{3}, \\textcolor{orange}{1} \\}$ since $\\{ \\textcolor{blue}{2}, \\textcolor{orange}{0} \\} \\in p_3$", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tdNzunhZ8p", "reviewer": "Reviewer_ZuTu", "summary": "This paper investigates compositional generation in diffusion models through the lens of fuzzy logic. By building on theoretical foundations in score-based generative models, existing compositional operators, and fuzzy set theory, the authors proposes Dombi operator that addresses limitations of prior approaches and demonstrates improved stability. Experimental results on small diffusion models, Stable Diffusion 1.4, and protein synthesis further support the effectiveness of the method.", "questions": "- How does the Dombi operator affect inference-time efficiency when applied to large-scale diffusion models (e.g., Stable Diffusion)? Is the overhead negligible or significant?\n    - How does the method extend to models with stronger inherent compositional abilities (e.g., SDXL, SD3)? Do the benefits persist or diminish?\n- Were the images in Figure 3 generated with FKC correction? If so, how large is the performance gap between Dompi with and without FKC correction?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper is grounded in solid theoretical analysis, clearly identifying weaknesses in existing compositional operators for diffusion models.\n- The proposed Dombi operator demonstrates strong compositional performance (e.g., Figure 3) and provides a flexible framework for combining multiple operators.", "weaknesses": "- The presentation of Sections 2-5 would benefit from improved clarity: several terms are introduced without sufficient explanation, and there are noticeable typos that hinder reading.\n- Although the theoretical justification is strong, the empirical evaluation is relatively limited. Additional qualitative visualizations and ablation studies would help further substantiate the claims, particularly in the context of real-world diffusion models.\n- Some empirical results for SD1.4 are referenced as being in the supplementary material (in Appendix C.2), but no supplementary was provided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Csl9tjGepK", "reviewer": "Reviewer_ndFP", "summary": "The paper examines techniques to compose diffusion models with set theoretic operations such as unions and negations. They find inconsistencies in existing techniques, in particular for negation and complex combinations of operations. They then address these problems with Dombi operators which they derive rigorously from fuzzy logic. They show that their operators are more well behaved and allows for consistent composition and algebraic operations and show more stable sampling of their method on three datasets resulting in higher sample quality.", "questions": "- How sensitive is the lambda hyperparameter?\n- Did you try larger logical formulas? How robust is this?\n- How expensive is the sampling; does the integration become more expensive in particular with the mentioned oscillations, and does the construction of the composition incur any computational overhead?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- Composing diffusion models is extremely useful, in particular negation has many practical use cases.\n- The paper clearly demonstrates the shortcomings of existing approaches to motivate the need for their method. \n- The operators are rigorously derived from fuzzy logic, putting their method on firm mathematical grounds\n- The experimental results convincingly demonstrate the effectiveness of their method", "weaknesses": "- The presentation of the paper could be slightly improved. For example, the different colors in figure 2 are too similar, making it hard to read. - - It is also not clear which of the dark curves represents which probability distribution\n- Some variables seem to be undefined or fall out of nowhere, like the form of g(x) and s_c in definition 4.1 \n- The used lambda hyperparameter seems to very significantly between different experiments, raising the question how carefully this has to be tuned", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "51XlsbdbDs", "reviewer": "Reviewer_nPjQ", "summary": "This paper proposes Dombi operators as a principled way of implementing conjunction, disjunction, and negation operations for composing diffusion models. The Dombi operators seek to correct flaws in the standard implementations of the composition operations including bias, instability, and failure to obey compositional laws. The proposal is tested on image and molecular generation problems.", "questions": "Can you comment on connections to the Du/Liu variants of compositional operators as discussed in Weaknesses? Did you try any of these variants for comparison, or test whether they suffer from the identified failure cases?\n\nDiscussion around Definition 4.1: I wonder if there is any connection with parametrization-independence here. Bradley&Nakkiran25 (Appendix I) (https://arxiv.org/pdf/2502.04549) (and possibly other references) suggest that parametrization-independence may be a beneficial property for compositional operators (a property possessed by the CFG operator and the Du/Liu styles of product composition); they also show that all 1-homogeneous operators are independent of parametrization. The Dombi operators you obtain appear to be 1-homogeneous as well. Can you comment on potential connections with parametrization-independence and/or other reasons that 1-homogeneity may be desirable for these operators?\n\nMinor comments:\nL123 “important to note”?\nL142: CFG is usually defined as $p^1(x)^\\gamma p^2(x)^(1 - gamma) = p^2(x) (p^1(x)/p^2(x))^\\gamma$ — not sure what is meant here?\nL215 I think this is the first time $\\phi_c$ is mentioned; if so, not very well motivated/introduced \nL244 Without reading Appendix A, it’s not clear what $g$ is", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I appreciate the rigorous derivation of \"correct\" operators for conjunction, disjunction, and negation. The perspective is helpful for clarifying and unifying a somewhat-messy landscape in which different works use different implementations of compositional operators loosely tied to logical operators -- but which may not represent those logical operators very well, and also may not combine together (e.g. conjunction with negation) in logically consistent ways.", "weaknesses": "The analysis of the flaws in PoE/MoE methods, as well as the comparison of the Dombi operators to \"standard implementations\", make particular choices of implementation that are in some sense the \"naive\" ones rather than what I would consider the \"standard\" ones: for example, implementing conjunction as $p^1(x) p^2(x)$ and negation as $p^1(x) / p^2(x)^\\gamma$. Works (referenced in this paper but not discussed in great detail) such as Du23 and Liu22 implement conjunction not as a simple product but actually $p(x) \\prod_i p(x|c_i) / p(x)$ (for two distributions, $p(x|c_1)p(x|c_2) / p(x)$), and negation not as $p(x|c_1)/p(x|c_2)$ but $p(x) p(x|c_1) / p(x|c_2)$, where $p(x)$ is the unconditional. Also, CFG is essentially a negation $p(x|c)^\\gamma p(x)^{1 - \\gamma} = p(x) (p(x|c)/p(x))^\\gamma$. Choices like these work better in practice, and seem to be closer to the Dombi operators derived in Def 4.1. (For example the denominator in (8) is somewhat related to the unconditional). I also suspect that these implementations might not suffer from some of the failure cases (instability, failure to compose, etc.) I feel that these Du/Liu variant would be a valuable (fairer) baseline for comparison. (I still think there is theoretical value in the derivations of the Dombi forms even if they turn out to be similar to the Du/Liu variants, because to my knowledge the forms described in these earlier works were essentially discovered to work well empirically without much theoretical justification.)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)