Simple and Critical Iterative Denoising: A Recasting of Discrete Diffusion in Graph Generation
Authors: Yoann Boget
Venue: ICML 2025 poster (2025)
Keywords: Graph
molecules
generation
discrete
diffusion
denoising
flow
matching
generative
compouding
errors
iterative
critic
critical
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We propose a novel framework called Simple Iterative Denoising, which simplifies discrete diffusion and circumvents compounding denoising errors by removing dependencies on previous intermediate states in the noising process.

Abstract: Discrete Diffusion and Flow Matching models have significantly advanced generative modeling for discrete structures, including graphs. However, the dependencies of the noisy distributions across time of these models lead to error accumulation and propagation during the reverse denoising process—a phenomenon known as \emph{compounding denoising errors}. To address this problem, we propose a novel framework called \emph{Simple Iterative Denoising}, which simplifies discrete diffusion and circumvents the issue by removing dependencies on previous intermediate states in the noising process. 
Additionally, we enhance our model by incorporating a \emph{Critic}, which during generation selectively retains or corrupts elements in an instance based on their likelihood under the data distribution. 
Our empirical evaluations demonstrate that the proposed method significantly outperforms existing discrete diffusion baselines in graph generation tasks.

Simple and Critical Iterative Denoising:
A Recasting of Discrete Diffusion in Graph Generation
Yoann Boget 2
Abstract
diffusion, where the noising process progressively masks elements of an instance. The problem has been observed in discrete sequence modeling, where techniques such as corrector sampling have been proposed to mitigate it.
Discrete Diffusson and Flow Matching models have significantly advanced generative modeling for discrete structures, including graphs. However, the dependencies between intermediate noisy states lead to error accumulation and propaga- tion during the reverse denoising process a phe- nomenon known compounding denoising er- rors. To address this problem, we propose novel framework called Simple Iterative Denois- ing. which simplifies discrete diffusion and cir- cumvents the issue by assuming conditional in- dependence between intermediate states. Addi- tionally, we enhance our model by incorporating a Critic. During generation, the Critic selectively re- tains or corrupts elements in an instance based on their likelihood under the data distribution. Our empirical evaluations demonstrate that the pro- posed method significantly outperforms existing discrete diffusion baselines in graph generation tasks.
100
Validity (%)
Discrete Diffusion (Marg.)
Simple Iterative Denoising (Marg.) Ours Critical Iterative Denolsing (Mask) Ours
16
32
64
126
256
512
NFC
Figure 1: Validity rate (without correction) of generated molecules trained on Zinc250k as function of the Num- ber of Function Evaluations (NFE) for three models: Dis- crete Diffusion (baseline). Simple Iterative Denoising (ours), and Critical Iterative Denoising (ours).
1. Introduction
Denoising models such as Discrete Diffusion and Discrete Flow Matching have significantly advanced generative mod- eling for discrete structures (Austin al., 2021; Campbell al., 2022; 2024; Gat et al., 2024), including graphs (Hae- feli et al., 2022; Vignac al., 2023). Despite their success, these models suffer from a key limitation caused by the dependencies between intermediate noisy states in the nois- ing and denoising processes. Errors introduced early in the process accumulate and propagate, degrading generative performance. This issuc is particularly pronounced in mask
In graph modeling, marginal distributions over node and edge attributes have proven to be a more suitable noise dis- tribution than the mask distribution, as corroborated by our experiments. However, we show both theoretically and em- pirically that compounding denoising errors also affects dis- crete diffusion and flow matching models when the marginal distributions are used as the noise.
To address this challenge, we introduce a novel framework called Iterative Denoising, which simplifies discrete diffu- sion by assuming that intermediate noisy states depend only on the clean data, making them conditionally independent of one another. By removing direct dependence partially denoised instances from previous steps, our framework facil- itates error correction and substantially improves generative performance.
Data Mining and Machine Learning group. School for Busi- ness Administration (HEGGGeneva)) University of Applied Sci- ences and Arts of Western Switzerland *Computer Science De- partment, University Geneva, Switzerland, Correspondence to: Yoann Boget <yo.nn.oogcceeeeeecc heege.hssg...>...
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Furthermore, we show that our model can be interpreted as selectively corrupting certain elements while leaving others

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Simple and Critical Iterative Denoising
untouched. Building on this insight, we introduce Critic, which modulates the corruption probability of each element based on the element's 's likelihood under the data distribution. We provide theoretical motivation for our approach and demonstrate empirically that the method further enhances models' performance.
integers in an annotation vector [dx]", where d is the number of node types.
To facilitate readability and simplify the transferability of the method to other discrete modalities, such as sequence modeling, we use 2 to denote a single element. In graph modeling, z may refer to either a node or an edge attribute when the formulation applies to both xx(9) and a(+-), The corresponding capital symbol Z represents the entire in- stance, such as graph G. For simplicity and consisteney with common practice, we use p(x) to denote the Probabil- ity Mass Function (PMF) P(X = a data point element is represented by the Dirac delta dis- tribution, denoted as 65, (z), with all the mass concentrated on 21. We sometimes represent the univariate categorical distribution over the variable p(z) as a vector z, where the i-th component 2; denotes the probability that belongs to the category indexed by i.
Our empirical evaluations highlight the effectiveness of the proposed framework, demonstrating superior performance over existing discrete diffusion baselines in graph genera- tion tasks. Notably, our method tackles the challenge of generating valid molecular structures. In Figure 1, we see that our method needs only a small number of denoising steps to reach almost 100% molecule validity, while the standard Discrete Diffusion tops at 80%.
z). The distribution of
Our key contributions are follows:
We identify compounding denoising errors key limitation in discrete diffusion across various noise distributions, not just masking, as previously reported.
We refer to mask distribution as the distribution in which all the probability mass concentrated on an additional syn- thetic attribute labeled as Mask, denoted by 6 This distribution sometimes referred to as the absorbing-stee -taate distribution (Austin et al., 2021). We define mask diffusiio as the class of diffusion models that employ this distribution as noise. We use to refer to the noising distributions and p to the denoising ones.
To address this issue, we propose a novel, simple, and effective framework: Simple Iterative Denoising (SID).
We extend SID with a novel Critic mechanism that adjusts re-masking probabilities based on the predicted clean sample.
2.2. Discrete Diffusion Models
Empirically, we show that our approach outperforms standard discrete denoising in graph generation tasks, with the Critic providing additional gains, especially in low-NFE settings.
In the forward process of diffusion models (Song & Ermon, 2019; Ho et al., 2020; Song et al., 2021), each element transitions independently from the data distribution g (z) 82, (2) at time f = 1 to a noise distribution (2) at time = O, which contains (or tends contain) no information about the original data.
Our code is available at: github. www.yoooget/steeoeeoooooooooooee
2. Backkrrund
The forward noising process is expressed under Markovian assumption as:
Graph diffusion models are powerful methods for generat- ing discrete graph structures, such as molecules. However, existing approaches, including Discrete Diffusion Models and Discrete Flow Matching (DFM), face challenges due to the temporal dependencies of the noising process. These dependeneies cause error propagation and accumulation, ultimately degrading generative performance. This section introduces core concepts, discusses these limitations, and motivates our approach.
411(2 21) = II gmi+ta artaeds
(1)
rer
where - (1 A....., + t.
The continuous-time case is described by taking lim gqn (z 21), which transforms the product into geometric integral.
For categorical (non-ordinal) discrete data, popular family of noising distributions is given by:
2.1. Notation
We define graph set of nodes and edges, denoted by g = (V,E). A graph is represented by its adjacency matrix A [dA + 1|mxn,, where n = [V). and [d + 1] (1....,dy da + 1}. with d representing the number of edge types and the additional label corresponding to the absence of an edge. Node attributes, if present, are encoded as
9 9--,,1(22 21) = (1- (,) 8,)&, (2) + 3, go(2).
(2)
This formulation includes forward processes leading to a uniform noise (go(2) = Zzz Vz), mask noise (qu(2) = SMack or marginal noise (go(=) = my, where m
2

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Simple and Critical Iterative Denoising
corresponds the empirical proportion of elements with attribute in the dataset.
The compounding denoising error is particularly evident in mask diffusion, where unmasked elements cannot be masked again, the conditional distribution collapses to Dirac delta:
Under this formulation, the forward process admits a con- venient closed-fone form solution to compute (z] 21) effi- ciently:
P/ Z,) = (2)) vs 2 t if 2, + Mask.
(4)
1 1 21) agdz (z) + (1 (z))
While the issue is particularly pronounced in mask diffusion, we argue that it also affects other noising distributions qu such as the uniform and marginal distributions.
where 04 1l B.).
This noising process has been expressed using transition ma- trices in prior works, including Austin et al. (2021), Camp- bell et al. (2022), and Vignac et al. (2023) for graph-based formulations. We note that standard instantiations of DFM for categorical data with independent coupling (Gat et al., 2024) are equivalent to continuous- catinuuus--tmme discrete diffusion. Therefore, the statements about discrete diffusion in the fol- lowing also apply these standard instantiations of DFM.
In these cases, the compounding denoising error is some- what less severe all elements can change at each denoising step, allowing for corrections. However, due to the direct dependence of 2₃ (red arrow in Figure ability of correction is low, and the risk of error propagation and accumulation remains significant. Specifically, Gat et al. (2024) show that a denoising step in discrete diffusion can expressed as:
(a)) the prob-
By learning backward process (e Z,) for t + At and sampling iteratively from it, we generate data samples starting from the noise distribution.
PLt./e(22 2e) = 8b (z)+ A a [pije(z] 2,) - 82, (z)]
21
Z
Zt
ZO
Data
Noise
where di the or time-derivative. The second term on the right-hand side represents the probability of modifying an element in denoising step of size A, The scaling factor AIT- in particular, imposes strong constraint on the probability of modifying an element. Consequently, pounding denoising errors are not limited to mask diffusion, but also affects other common noise distributions, such as uniform and marginal distributions.
(a) Discrete Diffusion.
21
Zz
%
20
Data
Noise
(b) Iterative Denoising.
2.4. Motivation
We argue that direct dependence on previous intermediate states in existing models hinders the generative performance of discrete diffusion models for graph generation. These models still struggle to consistently generate graphs with desired structural properties, such valid molecular graphs planar graphs.
Figure 2: Noising Graphical Models in (a) Discrete Dif- fusion: 2t (z]o(2 2,); and (b) Iterative Denoising: 2, (z =1)
2.3. Denoising Dependencies and Compounding Error
In this work, we propose simple yet effective framework that mitigates compounding denoising errors by removing dependencies on previous intermediate states in the noising process. Unlike standard DDM and DFM, which rely on the Markovian assumption, our Iterative Denoising approach (Sec. 3) assumes conditional independence from prior noisy states, alleviating the problem of error accumulation.
At the beginning of the discrete diffusion backward process, Z, contains limited information, resulting in a high-entropy distribution 2 [2] Zz), from which the elements 2, Z,, independently sampled. At time s, the joint distribution from the backward process:
1 (Z)) 2,)
II (1 Ze) ==2
Additionally, we introduce Critic-Guided Sampling pro- cedure (Sec. 4), which prioritizes re-noising elements with lower probabilities under the data distribution, further im- proving generative performance, especially in molecular graph generation.
may deviate significantly from the forward distribution due to early-stage denoising errors. This leads to error accumu- lation, a phenomenon known as the compounding denoising error (Lezama et al., 2023).
3

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Simple and Critical Iterative Denoising
2.5. Related works
Furthermore, our approach generalizes beyond masking support various noising distributions. Nonetheless, their pirical results further support the effectiveness of the Simple Iterative Denoising framework.
Our work primarily relates two areas of research: methods that address compounding denoising errors and generative models for graphs. In the following, we review relevant contributions in both fields.
2.5.2. GRAPH GENERATIVE MODELS
2.5.1. ADDRESSING COMPOUNDING DENOISING ERRORS
A key advantage of discrete denoising methods, including discrete diffusion, discrete flow matching, and our iterative denoising, is that they do not depend on a specific element ordering. This property offers flexible framework for sequence infilling and generation of unordered structures such as sets and graphs. In this work, we focus on the application of denoising models for graph generation.
Discrete diffusion approaches to mitigating compound- ing denoising errors follow two main strategies: corrector sampling and planning. Corrector sampling (also known as predictor-corrector sampling) (Campbell et al., 2024; Lezama et al., 2023) involves alternating denoising (back- ward) steps with noising (forward) steps to improve conver- gence. Recent work (Gat et al., 2024) integrates denoising and corrector sampling by simultaneously performing backward and 1 forward steps. In the context of mask diffusion, this method allows for the correction of previ- ously sampled elements. In Section 3, we show that our Simple Iterative Denoising process is formally equivalent to corrector sampling with maximal corrector steps. However, because elements are randomly re-noised, the efficiency of this approach remains limited, Extending this framework, our Critical Iterative Denoising (CID) introduces modulated re-masking probabilities, effectively acting as a probabilistic planner for re-masking decisions. Critically, unlike prior work, CID operates directly on fully denoised predictions.
Recent denoising generative models for graphs leverage the order- order-agnostic nature of denoising processes to maintain permutation equivariance. Some approaches assume contin- uous diffusion spaces such as score-based models (Yang et al., 2019; Jo et al., 2022), diffusion bridges (Jo al., 2024), and flow matching (Eijkelboom et al., 2024) but this assumption breaks the discrete nature of graph struc- tures during noising. Discrete diffusion models, whether in discrete time (Haefeli al., 2022: Vignac et al., 2023) or continuous time (Xu al., 2024), and discrete flow match- ing models, have shown strong performance graph gener- ation. However, these models are affected by compounding denoising errors. Our approach mitigates this issue, lead- ing to improved generative performance. A broader review of graph generative models, including non-equivariant ap- proaches, is provided in Appendix C.
Planning, as defined by Liu et al. (2025), involves select- ing which elements to denoise (i.e., unmask) at each iter- ation. This selection can be guided by confidence scores from model outputs (Zhao et al.. 2024b; Zheng al., 2024) by dedicated planner (Liu et al., 2025; Kim et al., 2025#) Outside discrete diffusion, confidence-based un- masking has been explored by Chang et al. (2022). Notably, planning does not influence the model's capacity to revise previously sampled elements. Using mask diffusion, Peng et al. (2025*) mitigate this issue by jointly deciding which masked elements unmask and which unmasked elements to remask. In contrast, our Simple Iterative Denoising ap- proach removes the need to explicitly plan a denoising order by predicting all elements simultaneously at each step.
11
Zs
Zt
(a) Discrete Diffusion
Zs
Ze
(b) Simple Iterative Denoising
While our method operates in a discrete setting, it relates to the general degrade-and and-r-oorr- strategy developed for im- age generation by Bansal et al. (2023). concurrent work, Nie et al. (2025*) introduce large language model lever- aging a mask denoising approach similar to ours. However, whereas they only briefly outline their sampling procedure, we provide a comprehensive theoretical foundation, ana- lytical motivations, and empirical evidence for the method.
Figure 3: Denoising Graphical Model for a single denois-
ing step in (a) Discrete Diffusion: &, ~ Paje(z Z,) = Σ., Pe|Le(> 21, 2))Pe(( Zt); and in (b) Simple It-
erative Denoising 2, ~ Peje(z Z,) = En, daj1 21 21)Ptf(= Zz).
3. Simple Iterative Denoising
References marked with denote concurrent work, meaning they were published after this article submitted.
In this section, we introduce Simple Iterative Denoising (SID), a simple yet effective generative framework for mod-

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Simple and Critical Iterative Denoising
eling discrete-state graphs. This approach simplifies existing denoising models, such as DDM and DFM, while enhancing generative performance. SID serves as the foundation for Critical Iterative Denoising (CID), which further improves the generative performance.
Despite this assumption, we note that the distribution at time in our method is identical to that of discrete diffusion, as formalized in the following proposition:
Proposition 3.2. Given a noise distribution qu and a sched- uler a(I), the distribution qal g (z 21) defined by our noising procedure (Equation 6) is equal to the distribution 4ej1 apPP (= =1) obtained by the discrete forward process distribution defined in Equation 1.
Similar to DDM, SID consists of both noising and denoising operations. The generative sampling procedure involves iteratively applying small denoising steps, enabling effective graph generation.
Notably, if we have access to a pre-trained DDM or DFM
Proof. See Appendix A.1
denoiser, i.c., parameterized model predicting PL (z Zt), the SID method is training-free free. Table 3 summarizes the key differences between SID, DDM, and DFM. In the following, we present the method in detail.
The proposition follows from the fact that our noising proce- dure corresponds to the closed-form expression (Eq. 3) used in discrete diffusion to sample efficiently from quj1 21). However, make no assumption that the noisy distribution results from a diffusion process.
3.1. Noising
We define a noising process that independently acts on each element of an instance as follows:
3.2. Denoising
Denoising progressively refines noisy instances by learning the distribution Paja [ Zz), where t + At. By leverag- ing Assumption 3. 1, the denoising process is significantly simplified compared to discrete diffusion, as formalized in the following proposition:
q (2 21) = 52 (z) + (1- a1--a))que (z)),
(6)
where (0(=) represents the noise distribution 6z, (z) is the data distribution, and ar is a non-decreasing scheduling parameter in [0, 1]. The scheduler a₁, which defines the noise level, depends on continuous time parameter, which also takes values in [0, 1].
Proposition 3.3. Given the noising process defined in Equa- tion 6 and Assumption 3.1, the denoising process is ex- pressed as:
Unlike standard diffusion models, which compute noise distributions via a Markov process (Eq. 1) such that (sl+AA. (z 21) 9rj++,, =+ti)), our inter- mediate noisy distributions are directly parameterized as a mixture of the noise and the data distribution. Thus, we do not assume any dependencies in the noising process other than the dependency on the original data point 21 and the noise level. This assumption introduces significant simplification over DDM and DFM, leading to beneficial implications for the SID denoising process. In Figure 2. we juxtapose the graphical models of our method and DDM at the noising phase. More formally we assume conditional independence of the noisy distributions across time given 21:
Pajo(= Zt) asPij(<]] Z,) + (1 ax)9a(z),
(8)
Proof. See Appendix A.3.
Under Assumption 3.1, the denoising process factorizes as:
Paje(2 Zz) = Σ Pio(21 Zz)4,((2 21).
(9)
Thus, we interpret the denoising as a two-step process: 1. Predict clean instance from the noisy data, P1j{ Z,), 2. Re-noise the predicted clean instance, (aj1 21). The key difference from the standard DDM model is that 2, in SID directly depends only on the predicted clean instance, but not on 2, In Figure 3 we juxtapose the denoising graphical models of DDM and SID. The fact that 2, is generated only from the predicted clean instance acts as a barrier to error propagation. This factorization will also prove useful in Section 4.
Assumption 3.1. The intermediate noisy distribution at time is conditionally independent from other intermediate states such that:
(1 21) q: 1= | 22,
Vt
As result, our noising procedure is not a diffusion process in the traditional sense, the state at time depends only the initial state and not on the noising trajectory or previous intermediate states. Under this assumption, the denoising phase does not suffer from error propagation and accumu- lation caused by the direct dependencies to previous states inherent in standard DDM and DFM (see Section 3.2).
At inference time, we define the Number of Function Evalu- ations (NFE), denoted as T, as a hyperparameter. The time step size is set to A, = 1/TT. The whole denoising process consists of iterating T times the denoising step defined in Equation
5

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Simple and Critical Iterative Denoising
3.2.1. CORRECTOR SAMPLING INTERPRETATION
4. Critical Iterative Denoising
The denoising process can also be viewed through the lens of discrete diffusion corrector sampling. Defining discrete diffusion corrector sampling step as applying kåc backward steps and (k 1) A forward steps, and noticing that the maximal size for corrector step at time is KA, - 4, we state the following proposition:
This section introduces Critical Iterative Denoising (CID), method designed to improve Simple Iterative Denoising.
As discussed in Section Mask SID can be viewed sequence of denoising steps comprising: (1) unmasking ele- ments of an instance based on the denoiser's prediction and (2) re-masking random subset of the instance's elements.
Proposition 3.4. Simple Iterative Denoising step, as de- scribed in Equation 8, equivalent to a discrete diffusion corrector sampling with maximal corrector step sizes.
However, in Mask SID, all elements are re- re-masked with the same probability. This approach is suboptimal, as not all elements are equally likely under the data distribution after the unmasking step.
CID addresses this limitation by dynamically adjusting the re-noising probability during sampling. Specifically, CID increases the re-noising probability for predicted denoised elements overrepresented under the data distribution and decreases for those underrepresented. CID serves two goals: (1) Reducing error propagation by resampling el- ements that are likely out of the data distribution, and (2) Accelerating inference by decreasing the number of function evaluations required during sampling. As shown in Figure 1, CID achieves a state-of-the-ard validity rate on Zinc250k with only a few dozen of denoising steps.
Proof. See Appendix A.2.
Consequently, the denoising process inherits the properties of the backward process in discrete diffusion; by iteratively sampling Ps (= Ze) with sufficiently small A., the denois- ing process gradually transitions from the noise distribution to the data distribution. Our mask SID) also inherits the time-independeece property of mask denoisers shown by Ou et al. (2025) and Sahoo et al. (2024).
3.3. Parametrization and Learning
4.1. Preliminaries
As in discrete diffusion, the conditional probability (2) Z,) is intractable. We model this distribution by parametriz- ing a neural network fe Te( a.) referred to as the denoiser. We use Graph Neural Networks (GNNs) to enforce equivari- ance to node permutations, thereby preserving the structural invariance of graphs.
Before describing the method, we recast the definition of the noise distributions by introducing a Bernoulli random variable a,, whose mean is given by the scheduling param- eter a from Equation 6; a, indicates whether an element has been corrupted (a: 0) or not (a, =1). As with z, we use the capital letter A to denote the set of indicators for all elements in an instance. This formulation allows us to interpret the noising process as selectively corrupting some elements while leaving others untouched. Through the random variables a,, we can identify which elements 2; have been noised. This identification would otherwise be impossible for any noise distribution sharing support with the data distribution (i.e., all practical noise distributions except the mask distribution).
Since the training objective matches that of DDM and DFM, can adopt any of the training criteria proposed for these models to train fo Z2, ox). Following Digress (Vignac et al., 2023), we minimize the weighted negative log- log-likelihood in our experiments:
L -= G--p,p.t--210111,---- (a (22) (1
Given the random variable a, ~ Pa. (a) = Bernoulli(a;aa). we rewrite Equation as:
(10)
(=)()))))) Pa. (a)o (2) + Pa, (a)) (11)
log(po(1)) (2) [())]
Similarly, we write the denoising equation as:
log(po (e) [.
(1 7)
P1μ(2 Ar) a. (=)+
Ze, As) (12)
Intuitively, when the element is not corrupted (ar = 1), it remains unchanged (6z, (2)): when a, = we predict it using the denoiser. For simplicity, we define Posa(2) := 8z, (2) and Prrea (2) == (= Zt, At).
where 7 is a weighting factor between nodes and edges. We use y = n2(nn + m). At inference time, Simple Iterative Denoising can leverage any pre-trained DDM model.
6

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Simple and Critical Iterative Denoising
In principle, our method supports any noise distribution and produces pairs (ze; at), which indicate whether the element & was sampled from the noise distribution or corresponds to the original clean element. However, denoising using the pair (zx, dz) reduces denoising models with any noise distribution (e.g., marginal, uniform) to a mask denoising model.
Lemma 4.3. Pfanaa (211e) Ppred Eije), the optimal Critic's
noising probability 02 is smaller than the schedule's es noising rate Ckt, that is:
Pdina (21te) > Ppread
Or 07.
(16)
Conversely, if Pdtto < (1)) the optimal Critic's
noising probability ä is larger than the schedule's noising rate dzy
To see this, note that the corruption indicator at acts as mask, with de = indicating masked element. Defin- ing the set of uncorrupted elements as Z = { E Ze a, = 1}. and its complement as Z the masked denoiser can be expressed as P" 1/ (= Z,,A,) Since Z carries no information about Z1 it follows that: (f A.)
Paato (21je) < Pprea((11e)) = or < aā.
(17)
Proof. See Appendix A.5
Additionally, if (11ee = 0, then εnjt will not be masked (a; = 0). Conversely, if Pdata (21je) 0, the element is out of distribution and will be masked with probability a₁ 1.
(= 7 A1). Because Critical Iterative Denoising (CID) explicitly requires A,, and any noise distribution under this configuration collapses to the Mask Simple Iterative Denois- ing (Mask SID) model, we always use Mask SID within CID.
Thanks to Theorem 4.1 1 and the following lemmas, the Critic steers the denoised element toward the data distribution, by re-noising with higher probability those elements that are overrepresented under the data distribution, while preserv- ing underrepresented ones.
4.2. Iterative Denoising with a Critic
We now describe the Critic, which predicts the corruption state of graph elements during denoising. Let 21j= ~ PiJe (= Zr Ar) denote denoised element at time t. CID trains Critic C to estimate & (a (jj)). the probability that 2 originates from Pdata rather than Ppred- The Critic is trained by minimizing the negative logeliiel of the corruption indicators aj produced during the noising phase (one per element and time step):
4.3. Implementation and Sampling
Since Pa, Pa.(a) (Equation 11) does not depend on Znp0 Eg 2 p(a Zaj() Pa. (a) = az, we actually cast the Critic as a predictor of the residual logit with respect the true at:
(5)
po(a
off az)
(2)++ to (ax)),
(18)
L. = -E, 2111 Σ loggoo(a 221e),
(13)
where a is the sigmoid function and fo (21je oz) is a GNN. The Critic operates on the outputs of a fixed denoiser and is trained post hoc, requiring no retraining of the denoiser itself. We leverage any denoiser implementing mask diffusion and no retraining is needed.
During inference, à parametrizes a Bernoulli distribution Pa, (at) = Bernoulli(a: a; âe) which determines the noising probability àē Recall that a; is the scheduler's noise rate time step i. Thus, the Critic's output can be interpreted as an adaptive, element-wing noise schedule.
During inference, we are interested in ár+s, rather than âg- We therefore use the approximation âc+s. x ( + +o (a(+t))))
The following theorem characterizes the optimal critic:
Theorem 4.1. The optimal Critic C* is:
In summary, each denoising step involves: 1. sampling 21ja from the Denoiser, 2. computing â,+2, via the Critic, 3. re-noising using Equation 6, with a replaced by âu+
(.Pámna@111)
(211)) aePaonaa (ePdona(2111) (1 at (pred(()))))
(14)
Proof. See Appendix A
5. Evaluation
From Theorem 4.1, two lemmas follow:
We evaluate our model on both molecular and synthetic graph datasets.
Lemma 4.2. If (dane(211)) PPrre((2 t.). the optimal 6; coincide with the true Ct2, that is:
For molecular data, we use the QM9 and Z INC2500k datasets. The QM9 dataset contains 133,885 molecules with up to 9 atoms of 4 types, while the 2INC25CK dataset consists of 250,000 molecular graphs with up to 38 heavy atoms of 9 types. For generic graphs, we run experiments on
Paata (21je) Ppreed aj as
(15)
Proof. The lemma follows directly from Theorem 4.1. LI
7

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Simple and Critical Iterative Denoising
the Planar and Stochastic Block Model (SBM) datasets. Both contain 200 unattributed graphs with 64 nodes and up to 200 nodes, respectively. Visualizations of generated molecules and graphs are available in Appendix G.
Table 1: ZINC250k.
Molecule generation results on QMB and
QM9
ZINC250K
MODEL
VAL.t
NPK.
FCD4
VAL.
NPK|
FCD
GDSS
95.72
3.3
2.90
97.01
19.5
14.65
Our objective is to evaluate the proposed method through direct comparison with Discrete Diffusion Models (DDMs).
DRUM
99.69
0.2
0.11
98.65
1.5
2.25
DIGRESS
98.19
0.3
0.10
94.99
2.1
3.48
To ensure a rigorous comparison, we train two denoisers, one using the marginal distribution and the other using the mask distribution. We use these denoisers to compare the two sampling procedures: Discrete Diffusion and Simple Iterative Denoising. Additionally, we train a Critic for the model using the mask distribution. Specifically, the Marginal DDM and Marginal SID share the same denoiser, while the Mask DDM, Mask SID, and Mask CID use another shared denoiser with identical architecture and pa- rameters.
MARG. DDM
95.73
1.92
1.09
80.40
12.96
8.50
MASK DDM
48.38
14.75
3.76
8.96
78.63
24.98
MARG. SID
99.67
1.04
0.50
99.50
2.06
2.01
MASK SID
96.43
1.40
1.80
93.85
11.08
9.05
MASK CID
99.92
1.40
1.76
99.97
2.26
3,46
(DDM) counterparts using identical denoisers both datasets and this across both noise distributions: marginal and mask. Furthermore, the results confirm our analysis of compounding denoising error, particularly affecting mask- based discrete diffusion, which performs poorly across both datasets. SID and CID address the compounding error successfully. Our experiments confirm that the marginal distribution is more effective noise distribution in graph modeling compared to the mask distribution. While com- pounding denoising errors are less severe with the marginal distribution, SID still consistently outperforms marginal DDM baselines by a wide margin. Finally, we observe that the Critic further improves the generative performance. Notably, on INC250K,, our Critical Iterative Denoising model reaches high-level validity rate, producing 50 times less invalid molecules than the best best-performing baseline, DruM.
This setup ensures a controlled and fair ablation study, iso- lating the impact of our iterative denoising approach while maintaining identical model architectures and training con- ditions across all comparisons. We provide the technical experimental details and implementation in Appendix E.
We report baseline results from various diffusion models; GDSS (Jo et al., 2022) is a continuous diffusion model, DruM (Jo et al., 2024) is a diffusion bridge model, and Di- Gress (Vignac et al., 2023) is a Discrete Diffusion model. DiGress uses the marginal distribution as noise. It differs from our Marginal DDM only by the network architecture, and hyperparameters. Baseline results are reported from Jo et al. (2024). Importantly, these baseline results were obtained using 1000 function evaluation steps during gener- ation, whereas we use only 500 steps in our experiments.
5.2. Generic Graphs
For generic graphs, we follow the evaluation protocol intro- duced by Martinkus et al. (2022) and adopted by Jo et al. (2022) and Jo et al. (2024), using an 80/20 train-test split with 20% of the training data reserved for validation.
5.1. Molecule Generation
We report the Frechet ChemNet Distance (FCD) (Preuer et al., 2018), which measures the similarity between gener- ated molecules and real molecules in chemical space, as well as the Neighborhood Subgraph Pairwise Distance Kernel (NSPDK- NPK) (Costa & Grave, 2010), which evaluates the similarity of their graph structures. Additionally, we report the proportion of chemically valid molecules (valid- ity) without any post-generation correction or resampling. Table presents the results. The best-performing models among our experiments are bolded. We also bold the best baseline if it outperforms ours. We generate 10,000 sam- ples, compared against 10,000 test molecules. Results are averaged over five independent sampling runs. Standard deviations and additional metrics, including uniqueness and novelty, are reported in Appendix F.
Table 2: Graph generation results on Planar and SBM datasets.
PLANAR
SBM
MODEL
SPECT.
V.U.N.1 SPECT.J
V.U.N.F
GDSS
37.0
12.8
DRUM
6.2
90
5.0
85
DIGRESS
10.6
75
40.0
74
MARG. DDM
83.57
0.0
11.82
0.0
MASK DDM
84.44
0.0
11.38
0.0
MARG. SID
7.62
91.3
5.93
63.5
MASK SID
8.72
67.0
15.05
17.5
MASK CID
6.40
66.0
11.94
19.0
We observe that our Simple Iterative Denoising (SID) mod- els consistently outperform their Discrete Diffusion Model
We evaluate graph similarity using the Spectral Maximum
Mean Discrepancy (Spect.), which compares the spectra of the graph Laplacian between generated and real graphs.
8

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Simple and Critical Iterative Denoising
Unlike other metrics that rely local structural properties, such node degrees clustering coefficients, the Spectral MMD captures graph structures at any level. Addition- ally, we report the proportion of Valid, Unique, and Novel graphs (V.U.N.). For the Planar dataset, a valid graph must be both planar and connected. For the Stochast ic Block Model dataset, validity indicates that the gener- ated graph is likely follow the block model distribution used to generate the training data (i.e., an intra-community edge density of 0.3 and an inter-community edge density of 0.005). Uniqueness represents the fraction of unique graphs among the generated samples, and novelty the fraction of unique graphs that do not appear in the training set. In practice, all generated graphs in our experiments were both unique and novel; thus, V.U.N. effectively measures validity. Table 2 reports the results. We average over five sampling runs, comparing the test set to a generated sample batch of equal size. Standard deviation and complementary metrics including MMDs based on degree, clustering coefficient, and orbits counts are reported in Appendix F.
the number of denoising steps needed to achieve high valid- ity, Mask CID reaching over 99% validity in just 32 steps. Notably, in the low-NFE regime (16 steps), Mask CID con- sistently outperforms all other models across all metrics, demonstrating its effectiveness in accelerating inference without sacrificing quality.
6. Conclusion
Discrete diffusion models suffer from the Compounding Denoising Error issue, which leads to error accumulation during sampling and negatively affects their performance. We address this issue by introducing an assumption of con- ditional independence between intermediate noisy states, Thus, the resulting method does not rely on a diffusion pro- cess. We further improve upon our method, and introduce a Critic, which steers the distribution of generated elements towards the data distribution, improving sample quality.
Our experimental results demonstrate the effectiveness of our method. Our Simple Iterative Denoising models con- sistently outperform their corresponding discrete diffusion models. Moreover, our mask model with Critic system- atically improves generative performance over the corre- sponding Simple Iterative Denoising mask model without the Critic.
The results on general graphs align with our findings on molecular graphs, further reinforcing our conclusions. Our Simple Iterative Denoising models consistently outperform their Discrete Diffusion counterparts, demonstrating their effectiveness across different graph types. Results also con- firm that mask-based models are not well suited for graph generation. In particular, on the SBM dataset, their results are barely significant due to their performances.
Additionally, we show that Critical Iterative Denoising significantly reduces the Number of Function Evaluations (NFE) required for sampling, making the generation process more efficient. While this work focuses on graph model- ing, future research should explore the applicability of our approach to other structured data domains.
While the Critic improves spectral MMD on the Planar dataset, its contribution is less pronounced for generic graphs than for molecular graphs. We hypothesize that this is due to deviations from the data distribution being less localized in generic graphs, making it more difficult to discriminate between graph elements. Using the marginal distribution, our Simple Iterative Denoising model achieves results significantly superior to the Discrete Diffusion base- line.
A Acknowledgments
am deeply grateful to Prof. Alexandros Kalousis, whose support has been invaluable far beyond what co-authorship could acknowledge.
acknowledge the financial support of the Swiss National Science Foundation within the LegoMol project (grant no. 207428). The computations were performed at the Univer- sity of Geneva on "Baobab", "Yggdrasil", and "Bamboo" " HPC clusters.
5.3. Ablation
We assess the impact of the Number of Function Evalua- tions (NFF)-i.e., the number of denoising steps during sampling-ov on model performance. Specifically, we investi- gate how the Critic affects the number of steps required to achieve high-quality generation.
Impact Statement
We generate graphs with varying NFEs in {16, 32, 64, 128, 512] and evaluate performance on the ZINC250K dataset for Marginal DDM, Marginal SID, and Mask CID. Validity is reported in Figure 1, while additional plots and full numerical results are provided in Appendix F.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Our findings indicate that the Critic substantially reduces
9

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Simple and Critical Iterative Denoising
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A. Proofs
A.1. Proposition 3.2: Noising and Diffusion Forward Process
Given a noise distribution, a data point, and a scheduler, the distribution ( defined by our noising procedure (Equation 6) is equal to the distribution 47f1 (z|z1) obtained by the discrete forward process distribution defined in (Equation I).
We adopt here the notation of Austin al. (2021), where the probabilities and probability transition are written as matrices, and start with the discrete time case.
Proof.
We have:
Q(jc+,, xQ
(19)
where Q: is transition probability matrix. In our case,
= +
(20)
where is an idempotent matrix, representing the noise distribution, the mask distribution being A le where em the one-hot vector indicating the mask and the marginal distribution being Ammy Im with m representing the marginal distribution.
Using the fact that 2 = A, and denoting a' B, we observe that:
11 Q: = II o I (1
11 0))
(21)
With a, = II a', we get: (0(2/2)) = 0n (2) + (1 az))(22)
For the continuous case, we remark that, a being smooth, we can turn the product into geometric integral.
A.2. Proposition 3.4
An Iterative Denoising step, described in Equation 8, is equivalent to corrector sampling discrete diffusion step with maximal corrector step.
Proof. Corrector sampling in discrete diffusion consists of applying k backward (denoising) steps, i.e., predicting Pt+*A,,b<=/a (=/2z)) and k - forward (noising step), 1ii+,s, (x)+ We call maximal corrector step a corrector sampling step such that - kA; = t. In this case, corrector sampling step becomes:
Σ an p(2(+,,22)
(22)
= (2)22) a142,)0002),
(23)
where the second equality follows from Proposition A.3.
A.3. Denoising
Given the random variable ae ~ Pa. (a) = Bernoulli(a; ax) (see Section 4.1), we have: Pat1 a.) = q0(2)) if = 0, and Paj1(2 (2/2,, a.) = dz, (z), otherwise.
14

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Simple and Critical Iterative Denoising
Po)r(2|(1)
Σ Paji (ani 2111/((1122
Σ Pan Pan(2|2)) Pan(elss)ppp(ee11k)
ΣΣ Pa. (a
(24)
Σ (a,52, (2) + go() -a,)) (11((21(2)
Σ Pıj((z 10)a (2) + D (1je(2112 [c) az)
aspitt(= + az)))(z)
A.4. Proposition 4.1: Optimal Critic
The optimal Critic is:
OssPaata (2118)
C (211))
(25)
arPatta (21je) + (1 at Ppreed (111)
Proof. The proof is inspired by Goodfellow et al. (2014). For single element, from Equation 13, we have:
C. -,,,
2n log (po(al211c)
-0t B2,, 1 Pauin log (po(alZinn (110)) - ot. 2, Pared log Pro(al 22le))
Σ arPaota (81/)) (po (a(aal211)) - - - ar) log (1 - (l-o((aa2 Z111))
(26)
21
(CPdata(@11p)) log (polal211)) + (1 (1-a))pped log (1 (a(a)2110)
For any vu, IR \ ()),0)) and y [0, 1], the function f(y) ulog(y) clog(1 v) reaches its maximum at
u++
Hence, Equation 26 reaches its minimum at:
@spdata
C ($11)) axPdata (E1je) (1 ar (111))
(27)
A.5. Lemma 4.3: Noising Rate
If Pdotaa (dota(211e) Ppred(211c), the optimal critic's noising rate 0. is smaller that the schedule's noising rate Br, that is:
(data(211c) (pre)(2111) az de,,
(28)
and conversely
Ddata(E1j6 < (pree(2111) = a4 > di-
(29)
15

---

Simple and Critical Iterative Denoising
Proof. From Theorem 4.1, have:
asPdata (21jc)
ā
(30)
acPdata (211e) + (1 ate Ppred
d₂ aePaata (211t) = arPdara (211e) + (1 as -a Ppred (211))
(31)
0n Pdato (21(e)
Pdata (Zite) + (ar² (ppred(211))
(32)
Pdata 21(e)
(33)
Pered &1le)
az
Hence:
a
Pdata(2111 > Ppred(2111)
> 1 = at
< az
(34)
ā
Pdata(2111) < Pored(211e) = at ae
(35)
B. Comparing Simple Iterative Denoising with Discrete Diffusion and Discrete Flow Matching
In Table 3, we compare Simple Iterative Denoising with Discrete Diffusion and Discrete Flow Matching. For Discrete Diffusion, we follow the framework of Austin et al. (2021), which uses uniform and mask (absorbing state) noise, and its extension to the marginal setting by Vignac al. (2023). For Discrete Flow Matching, we adopt the standard formulation from Gat et al. (2024), which uses independent coupling and convex interpolants. We the following shorthand notations: A, and D. A. A.T-a.
Table 3: Comparing Simple Iterative Denoising, Discrete Diffusion, and Discrete Flow Matching.
SIMPLE ITERATIVE DENOISING
DISCRETE DIFFUSION
DISCRETE FLOW MATCHING
NOISING
CONDITIONAL INDEP. GIVEN 21
MARKOVIAN
MARKOVIAN
Pqj (2
Ptpa
Pr|s(2|ze)
(2(2))
0482, (2) (1 (x)go(2)
a1d5, (2) (1 04 (o)2)
a+dz, (2) -x)))u(2)
(=)1())1))
a( la. q(aale1)
(1 1-)D),8, (2) + D,8,,
9(
(2))(2(2))
asPije(2 2,) + (1 - (x)
Σx, (2)r(2|2z, Zz, (z, 21)p1)(22
Ze)
82, (2) + D, (plj((2 2z)) - 6z, (2))
16

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Simple and Critical Iterative Denoising
Figure 4: Noising and Denoising Processes in Discrete Diffusion and Simple Iterative Denoising
0.00
220
C 0.00
C 0.00
L 0.00
C- 1.00
(a) Mask Discrete Diffusion
020
0.80
(b) Mask Simple Iterative Denoising
Unlike standard Discrete Diffusion, in Simple Iterative Denoising, an element that is masked (in forward) or unmasked (in backward) can be unmasked (in forward) or remasked (in backward). Note that for the sake of illustration, we use the absence of edge as an absorbing state. In practice, we use three different states for mask SID and mask DDM (mask, edge, non-edge).
C. Generative Graph Modeling: Related Works
A main challenge in generative graph modeling follows from the n! different ways to represent graphs due to node permutation. This has motivated two dominant families of approaches: sequential models, which generate graphs step-by- step, and equivariant models, which preserve equivariance to node permutation by design.
Sequential models construct graphs auto-regressively, adding nodes, edges, or substructures in predefined order (You al., 2018; Shi et al., 2020; Luo et al., 2021; Liao al., 2019; Kong al., 2023; Zhao al., 2024a). To limit the number of possible sequences representing single graph, many methods adopt Breadth-First Search (BFS) traversal strategy. In specific domains such as molecular graph generation, canonical formats like SMILES have been used to mitigate permutation et al., 2018; Kusner et al., 2017). However, general-purpos canonization strategies (Goyal et al., 2020) fail for large graphs (see experiments Bergmeister et al. (2024)). Hierarchical models that aggregate subgraphs (Jin al., 2018; 8; 2020) also fall into this category, though they rely on enumerating predefined vocabulary substructures and are thus limited constrained domains.
issues
Equivariant models aim to preserve permutation symmetry by producing the same output regardless of node ordering. These models have been instantiated across several generative paradigms, including GANs (Krawezuk et al., 2021; Martinkus et al., 2022), normalizing flows (Madhawa et al., 2019; Zang & Wang, 2020; Liu et al., 2019), and vector-quantized auto- encoders (Boget et al., 2024; Nguyen et al., 2024). More recently, equivariant denoising models in continuous spaces as score-based diffusion (Yang et al., 2019; et al., 2022), diffusion bridges (Jo et al., 2024), Flow Matching (Fijkelboom al., 2024) have significantly improved graph generation for small graphs. Despite their success, these continuous methods introduce mismatch between the continuous noise space and the inherently discrete structure of graphs.
To address this, several works have explored discrete denoising processes. Discrete diffusion models in both discrete time (Haefeli et al., 2022; Vignac et al., 2023) and continuous time (Xu et al., 2024) have demonstrated strong performance in generating small graphs. Discrete flow matching has also been adapted for graphs, with Qin et al. (2024a) showing promising results. However, these discrete models suffer from compounding denoising errors. Our proposed method directly tackles this issue, leading to improved generative performance.
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Simple and Critical Iterative Denoising
Scalability. A key limitation of current equivariant models is their reliance dense representations and pairwise computations, which hinder scalability to large graphs. To address this, recent works have proposed scalable denoising architectures and sampling strategies (Qin et al., 2024b; Chen et al., 2023; Karami, 2024; Bergmeister et al., 2024; Boget et al., 2025). Notably, Qin et al. (2024b) and Boget et al. (2025) propose methods that enable any equivariant model to scale to large graphs. While these methods could extend our iterative denoising framework to larger graphs, we leave this exploration for future research.
D. Models
D.1. GNNs architecture
Our denoisers are Graph Neural Network, inspired by the general, powerful, scalable (GPS) graph Transformer.
We implement single layer as:
()) E MPNN(X EE)),
(36)
X (1+1) MultiheadAAttenionn X x + X
(37)
E ((+1) E + E
(38)
where, (1) and E()) are the node and edge hidden representations after the (th layer. The Multihead Attention layer is the classical multi-head attention layer from Vaswani al. (2017). and MPNN is a Dessage-Passingg Neural Network layer described hereafter.
The MPNN operates each node and edge representations follow:
h/
ReLU(W) sn W ai + W Wedge e
(39)
J+1
LayerNorm( (h's
(40)
LayerNorm
x
fooce f(odc(()l.)) Σ
(41)
JEN())
where Wi sre W! Wing and Wi edge matrices of parameters and fnode, and fedge small neural networks.
The node hidden representation, i.e., the v,'s 's and the hidden representation of Snode are of dimensions dn, hyperparameter (see Table 4). The edge hidden representation, i.c., the ei,j's 's, h 's, and the hidden representation of fedge are of dimensions dr/4.
Inputs and Outputs In the input, concatenate the node attributes, extra features, and time step node features, copying graph-level information (c.g., time step graph size) to each node. The node and edge input vectors are then projected to their respective hidden dimensions, d, for nodes and d for edges.
Similarly, the outputs of the final layer are projected to their respective dimensions, d, for nodes and de for edges (or to a scalar in the case of the Critic). To enforce edge symmetry, we compute Ei tu. Finally, we ensure the outputs can be interpreted as probabilities by applying either softmax or sigmoid function, as appropriate.
D.2. hy pyprparraeeeers
As explained here above, the node hidden representation has size d, and the edge representation d,/4. We use d, = 64 with the QM9 dataset and dn 256 with all the other datasets.
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Simple and Critical Iterative Denoising
Table 4: Hyperparameters
MPNN layers
4
Layers in MLPs
3
Diffusion steps
500
Learning rate Optimizer
0.0002
Adam
Betas parameters for Adam Scheduler
(0.9, 0.999)
cosine (Nichol & Dhariwal, 2021)
D.3. Extra Features
Following common practice (Vignac al., 2023; Qin al., 2024a; Boget al., 2025), enhance the graph representation with synthetic extra node features. We use the following extra features: eigen features, graph size, molecular features (for molecular datasets such as QM9 and Z INC250K),, and cycle information (for the Planar datasets). All these features are concatenated to the input node attributes.
Spectral features We use the eigenvectors associated with the & lowest eigenvalues of the graph Laplacian. Additionally, we concatenate the corresponding k lowest eigenvalues to each node.
Graph size encoding the dataset, n Rrxx- This value is concatenated to all nodes in the graph.
The graph size is encoded the ratio between the size of the current graph and the largest graph in
Molecular features
For molecular datasets, we use the charge and valency of each atom as additional features.
Following Vignac et al. (2023), we count the number of cycles of size 3, 4, and 5 that each node is part of, and use these counts as features.
Cycles
E. Evaluation
E.1. Molecular Benchmark
For molecular graphs, adopt the evaluation procedure followed by Jo et al. (2024), from which took the baseline model results, and which was originally established in Jo et al. (2022).
Datasets The QM9 dataset (Wu et al., 2017) consists of 133,885 organic molecules with up to 9 heavy atoms, including carbon (C), oxygen (O), nitrogen (N), and fluorine (F). In contrast, the ZINC250K dataset (Irwin et al., 2012) contains 250,000 molecules with up 38 atoms spanning 9 element types: C, O. N, F, phosphorus (P), sulfur (S), chlorine (CI), bromine (Br), and iodine (I), Both datasets are divided into a test set (25,000 molecules), validation set (25,000 molecules), and a training set (the remaining molecules).
For our experiments, we preprocess the datasets following standard procedures (Jo et al., 2022: 2024). Molecules are kekulized using RDKit, and explicit hydrogen atoms are removed from the QM9 and ZINC250k datasets.
We evaluate the models using three metrics:
1. Validity: The percentage of chemically valid molecules among the generated samples, determined using RDKit's Sanitize function without applying post-hoc corrections, such as valency adjustments or edge resampling.
2. Fréchet ChemNet Distance (FCD) (Preuer et al., 2018): Measures the distance between feature distributions of generated molecules and test set molecules, using ChemNet to capture their chemical properties.
3. Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) MMD (Costa & Grave, 2010): Assesses the quality of graph structures by computing the maximum mean discrepancy (MMD) between the generated molecular graphs and those from the test set.
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Simple and Critical Iterative Denoising
E.2. Generic Graphs
We assess the quality of the generated graphs using three benchmark datasets from Martinkus et al. (2022).
Planar Graph Dataset: This dataset consists of 200 synthetie planar graphs, each containing 64 nodes. A generated graph is considered valid if it is connected and planar.
Stochastic Block Model (SBM) Dataset:
This dataset consists of 200 synthetic graphs generated using the Stochastic Block Model (SBM). The number of communities is randomly sampled between 2 and 5, and the number of nodes per community sampled between 20 and 40. The edge connection probabilities are set as follows:
Intra-communn aita--oomuuity edges: 0.3 (probability of edge existing within the same community).
Inter-community edges: 0.005 (probability of an edge existing between different communities).
A generated graph is considered valid if it satisfies the statistical test introduced in (Martinkus et al., 2022), which assess it corresponds to to the SBM structure.
We adopt the evaluation framework proposed by (Liao et al., 2019), utilizing total variation (TV) distance to measure the Maximum Mean Discrepancy (MMD). This approach is significantly more computationally efficient than using the Earth Mover's Distance (EMD) kemel,, particularly for large graphs.
Additionally, we employ the V.U.N. metric as introduced by Martinkus et al. (2022), which evaluates the proportion of valid, unique, and novel graphs among the generated samples. A graph is considered valid if it satisfies the dataset-speciffc structural properties described earlier.
F. Complementary Experimental Results
Results QM9 As additional metrics, report uniqueness, defined the fraction of unique molecules among the generated samples, and novelty, the fraction of unique molecules that are not present in the training dataset. Furthermore, all models achieve 100% validity with valency correction.
Table 5: Results on QM9
AVERAGE
VALID
UNIQUE
NOVEL
NSPDK
FCD
MARGINAL DDM
95, 73 + 0, 24 97, 52 1 0, 16 79, 55 + 0, 28 95, 0,28
1,922 0, 054
1,090 + 035
MARGINAL ID
99,67 0,00
72,
0,
0,041
0,504 010
MASK DDM
48,38 0,47 78,47
75,
25
+ 0,88
14,750
3,760 035
MASK ID
96, 63,43 0, 98, 02 04
86,
79 0,33
1,395
1,797 027
MASK CID
99, 92 = 0, 96, 93 0, 80,
0,22
1,402
1,757 # 0, 035
Results on ZINC250k As additional metrics, we report uniqueness, defined as the fraction of unique molecules among the generated samples, and novelty, the fraction of unique molecules that are not present in the training dataset. Furthermore, all models achieve 100% validity with valency correction.
Table 6: Results on ZINC250K
VALID
UNIQUE
NOVEL
NSPDK
FCD
MARGINAL DDM
80,40 + 0,31
95,93 + 0, 02
99,98 + 0, 03
12,96 + 0,25
8, 50 09 2,01
MARGINAL ID
99, 50 + 0,06
0,,8400000
99,97 00
2,06
MASK DDM
8.96
99,37
78,43 100,00
24,98
MASK ID
93, 85 + 0,
100,00
9,05 0,,0
MASK CID
99, = 0,001
99,98 0, 0, 01
99, 9. 0, 01
2,26
3, 46 4,
Results on Planar
We provide additionally the results using the degree, clustering and orbit MMD.
20

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Simple and Critical Iterative Denoising
Table 7: Results on P Planar
SPECTRAL
V.U.N.
DEGREE
CLUSTERING
ORBIT
MARGINAL DDM
83,57 56 1,32 2,77 1,72
0,0
63,83
3,
57
MARGINAL ID
91,3+4,1
1,23
163,40+31,80
19. 09.08 14
MASK DDM
25,0 0,0
20
34,
MASK ID
67,0+6,5
2,30
13,48
11,68 11,68 18,18
MASK CID
1,
5,5
0,
80,
Results on SBM
We provide additionally the results using the degree, clustering and orbit MMD.
Table 8: Results on S3M
SPECTRAL
V.U.N.
DEGREE
CLUSTERING
ORBIT
MARGINAL DDM
11,82 85
0,0 0,0
0,96 0, 73
66,66 6,26
32.37
MARGINAL ID
1,13
63,
2,,5
123 1,,4
MASK DDM
11,38
0,0
1,71
23,12
MASK ID
15,05
17,5+5,7
21,
55,
63 0,
127, 05 17,95
MASK CID
21,94 19,
43, 13 12.31
55, 06 2,29
92, 80 + 16,96
NFE Ablation ZINC250
We present here addition figures and tables ablating the effect of the NFE.
Diftusion (Marg.)- Baseline
Discrete (Marg.)- Baseline
Critical itcrative Denoising (Mask)
17.5
Critical Itcrative Denoising (Mask)
40
Iterative Denaising (Marg.)) Qurs
Iterative Denoising (Marg.)
15.0
NSPDK (20-1)
30
12.5
10.0
20
7.5
10
5.0
32
512
NHE
NHH
Figure 5: NSPDK and FCD as a function of the Number of Function Evaluations (NFE) for three models: Discrete Diffusion (baseline), Iterative Denoising (ours), and Critical Iterative Denoising (ours).
Table 9: Validity vs NFE
NFE
16
32
64
128
256
512
MASK ID
36,34
56,60
70.94
81,78
89,37
93,57
MASK DDM
9.35
8,84
9.05
9.14
9,06
8,49
MASK CID
92,51
99.17
99,78
99.91
99.98
99,96
MARG. DDM
57,31
69,33
76,80
78,09
80,15
80,37
MARG. ID
38.25
73,66
92,71
97.69
99,06
99,58
21

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Simple and Critical Iterative Denoising
Table 10:
NSPDK vs NFE
NFE
16
32
64
128
256
512
MASK ID
30,05
24,90
20,03
15,68
13,10
10,49
MASK DDM
78,27
79,66
77,94
79,59
79,11
77,97
MASK CID
11,10
5,66
3,55
2,64
2,30
2,26
MARG. DDM
47,22
27,35
20,23
15,74
13.75
13,46
MARG. ID
27.86
7,18
1,80
1,30
1,53
2,11
Table 11: FCD NFE
NFE
16
32
64
128
256
512
MARG, DDM
18,81
13,85
11,21
9,73
8,87
8,71
MARG ID
14,70
6,37
3,06
2,15
1,89
2,03
MASK DDM
25,23
25,35
24,73
25,09
24,96
24,69
MASK ID
15,80
14,16
12,43
10,80
9,80
9,01
MASK CID
9,42
6,63
5,05
4,09
3,77
3,40
22

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Simple and Critical Iterative Denoising
G. Visualizations
G.I. Molecular graphs
Figure 6: QM9
Generated molecules
Real molecules
HO.
HO
Figure 7: ZINC250K
Generated molecules
Real molecules
10tha
XAOOO Hop
soup
fo aays
omoot
23

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Simple and Critical Iterative Denoising
G.2. Generic Graphs
Figure 8: Planar
Generated graphs
Real graphs
Figure 9: SBM
Generated graphs
Real graphs
All generated graphs comes from the Marginal Iterative Denoising model.
24


Reviews:
[{"review_id": "BSOwxcoFsw", "reviewer": "Reviewer_VwCn", "summary": "The paper aims to address *Compounding Denoising Error* in discrete diffusion models by removing time dependency in the noising process. To enhance performance, it introduces a Critic that aligns generated samples with the data distribution. The authors provide relevant experimental results as supporting evidence.\n\n## update after rebuttal\n\nThank you for the authors' feedback. However, I have decided to retain my original score. I am not fully convinced by the proposed method's (1) novelty or (2) empirical performance.\n\nThe authors claim that one of the distinct components of the proposed method is that \"the reverse process is interpreted as predicting a clean instance and then re-noising it.\" However, this is only slightly different from the process described in Eq. (4) of D3PM [1] and Eq. (24) of [2]. These two prior works adopt a very similar design by first predicting clean data and then adding noise according to the time schedule, which is common in discrete diffusion models. Consequently, the methodological novelty seems limited to me.\nAdditionally, as pointed out by reviewer 4gxJ, the empirical study lacks comparisons with recent models, and the performance gains are neither consistent nor substantial compared to baselines.\n\nThe proposed method could potentially have significant impact; however, the current analysis in the manuscript does not provide sufficient evidence to support it. Therefore, I do not believe this paper qualifies for ICML acceptance.\n\nRef:\\\n[1] Austin J, Johnson DD, Ho J, Tarlow D, Van Den Berg R. Structured denoising diffusion models in discrete state-spaces. Advances in neural information processing systems. 2021 Dec 6;34:17981-93.\\\n[2] Gat I, Remez T, Shaul N, Kreuk F, Chen RT, Synnaeve G, Adi Y, Lipman Y. Discrete flow matching. *Advances in Neural Information Processing Systems*, 2024 Dec 16;37:133345-85.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their careful reading of our submission and for the thoughtful questions and comments. We believe these insights will substantially help us improve the quality and clarity of our paper.\n\n### 1. Time Dependency in the Noising Process\n\nWe at time used *time dependency* ambiguously in the original submission. Specifically, we occasionally referred to *time dependency* when we meant *dependent noise distributions across time*. We agree that this may have been misleading and will revise the camera-ready version accordingly .\n\nRegarding the difference with DFM, we remind that the standard instantiation of DFM with independent coupling is equivalent to continuous-time discrete diffusion, so that all explicit differences with DDM in Section 3 also hold for DFM. \n\nThe core difference between our approach and the Diffusion Flow Matching (DFM) model lies in **Assumption 3.1**. In particular, we assume *conditional independence* of the noisy variables across time steps, formalized in Eq. 7:\n\n$$\nq_{t|1}(z|z_1) = q_{t|1}(z|z_s, z_1) \\quad \\forall t \\neq s.\n$$\n\nThis assumption contrasts with the formulation in DFM with independent coupling [1], where the stochastic process is defined through the relation\n\n$$\nZ_s \\sim \\delta_{Z_t}(Z) + \\Delta_t u_t(Z_t)\n$$\n\n(Eq. 12 in [1]), thus inducing a direct dependency between $Z_s$ and $Z_t$, and violating Assumption 3.1.\n\nTo clarify further, the marginal noising distribution $q_{t|1}(z|z_1)$ (Eq. 6) is indeed equivalent to that of DFM when considering a single time step. However, the distributions $q_{s|t}(z|z_t)$ differ substantially, due to our assumption of conditional independence (See points 2 and 3).\n\n---\n\n### 2. Equation (7) and the Forward Process\n\nTherefore Equation 7 plays a key role in distinguishing our model from both DDM and DFM. While DDM employs a Markovian forward process such that\n\n$$\nq_{t|1}(z|z_s) = q_{t|1}(z|z_s, z_1),\n$$\n\nand DFM constructs a stochastic trajectory with explicit dependence between $Z_s$ and $Z_t$, our model, by contrast, does not define a probabilistic path or trajectory, but rather a serie of *conditionally independent* distributions given $z_1$.\n\nThis structural assumption leads to a simplified denoising process. Because $Z_s$ depends on $Z_t$ only through the intermediate variable $Z_1$, the reverse process interprets as predicting a clean instance and renoising it. This is operationalized through Eqs. 8 and 9 in our paper. The absence of a forward path avoids the compounding denoising errors that affects models such as DDM and DFM where $Z_s$ directly depends on $Z_t$.\n\n---\n\n### 3. Novelty of Section 3\n\nAs discussed above, **Assumption 3.1** (Eq. 7) leads to a distinct denoising process: we use Eq. 8, while DFM uses Eq. 5. Crucially, in DFM (as in DDM), the backward process involves direct dependencies between $Z_s$ and $Z_t$ (as expressed in Eq. 5), whereas in our model, this dependence is only indirect via $p_{1|t}(z|Z_t)$.\n\nThis decoupling directly addresses the issue of error accumulation in denoising, as discussed in Section 2. \n\nIn addition, the summary provided in the “Claim and Evidence” section of reviewer **whDB** accurately outlines the novel contributions of our paper. We provide further justifications in our response to reviewer **4gxJ** (Claim and Evidence - P. 1)\n\n---\n\n### 4. Similarity to Prior Work\n\nWe thank the reviewer for pointing out this related work. We will ensure it is incorporated in the camera-ready version.\n\nRegarding novelty: since the referenced work [2] builds upon DDM, all differences discussed above remain applicable. For a detailed comparison between our *Critic* and the *Planner* in [2], we refer to our response to reviewer **whDB**, who raised a similar question.\n\n---\n\n### 5. Comparison with Recent Empirical Studies\n\nAs elaborated in Appendix B (*Generative Graph Modeling: Related Works*), we classify generative models into *sequential (invariant)* and *equivariant* approaches. While some hybrid models attempt to merge these approaches, our work is *orthogonal* to these distinctions.\n\nOur implementation is based on an equivariant model to eliminate confounding effects from node ordering and to ensure a fair comparison with other equivariant baselines. Moreover, sequential models are particularly susceptible to overfitting, a concern we sought to avoid. In consequence, we selected comparable equivariant models as baselines.\n\nOn QM9, the results in [3] and [4] align with *DruM*, our primary equivariant baseline. On Zinc250k, although these models yield similar similar results to our *Marginal ID* on FCD and  NSPDK, our model produces *an order of magnitude fewer invalid graphs*, highlighting its robustness.\n\nWe will add the missing references in our literature review. However, we are not in favor of multiplying the baselines.\nFor a broader discussion on baselines, we refer to our detailed response to reviewer **4gxJ**\n(*Experimental Designs Or Analyses (Baselines)*).", "claims_and_evidence": "I have provided aggregated comments in this block. Please refer to them for the rebuttal discussion. Thanks.\n\n1. **Time Dependency in the Noising Process:**  \n   - The paper states that the noising process does not assume any time dependency apart from its dependence on the original data point \\$z_1$. Could the authors clarify how this aligns with Equation (6), given that the equation still contains a time-dependent coefficient?  \n   - Additionally, how does this approach differ from the discrete flow matching (DFM) formulation [1]?  \n\n2. **Equation (7) and the Forward Process:**  \n   - Could the authors clarify whether Equation (7) performs the same operation as DFM [1] in the forward process?  \n   - It seems to provide an analytic solution conditioned on the un-noised data—would the authors be able to confirm or elaborate on this?  \n\n3. **Novelty in Section 3:**  \n   - The model description in Section 3 appears to closely align with the DFM framework. Could the authors highlight any key differences or novel aspects introduced in this work?  \n\n4. **Similarity to Prior Work:**  \n   - There seems to be a notable similarity to the work presented in [2], especially its planner module. Could the authors clarify how their approach differentiates from or builds upon this prior research?  \n\n5. **Comparison with Recent Empirical Studies:**  \n   - How does the proposed approach compare empirically to recent advancements in graph generation, such as SwinGNN [3], Equivariant Denoisers [4], and PARD [5]?  \n   - These works have introduced new insights into permutation invariance and autoregressive diffusion models for graphs. Could the authors provide a comparative analysis or discuss potential advantages and limitations relative to these methods?  \n\n[1] Gat I, Remez T, Shaul N, Kreuk F, Chen RT, Synnaeve G, Adi Y, Lipman Y. Discrete flow matching. *Advances in Neural Information Processing Systems*, 2024 Dec 16;37:133345-85.  \n\n[2] Liu S, Nam J, Campbell A, Stärk H, Xu Y, Jaakkola T, Gómez-Bombarelli R. Think While You Generate: Discrete Diffusion with Planned Denoising. arXiv preprint arXiv:2410.06264. 2024 Oct 8.\n\n[3] Yan Q, Liang Z, Song Y, Liao R, Wang L. SwingNN: Rethinking permutation invariance in diffusion models for graph generation. *arXiv preprint arXiv:2307.01646*, 2023 Jul 4.  \n\n[4] Laabid N, Rissanen S, Heinonen M, Solin A, Garg V. Equivariant Denoisers Cannot Copy Graphs: Align Your Graph Diffusion Models. *The Thirteenth International Conference on Learning Representations*.  \n\n[5] Zhao L, Ding X, Akoglu L. PARD: Permutation-invariant autoregressive diffusion for graph generation. *arXiv preprint arXiv:2402.03687*, 2024 Feb 6.", "methods_and_evaluation_criteria": "Please see the comments above.", "theoretical_claims": "Please see the comments above.", "experimental_designs_or_analyses": "Please see the comments above.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Please see the comments above.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Please see the comments above.", "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": "0Jq7lKImOv", "reviewer": "Reviewer_whDB", "summary": "This manuscript observes that choices taken at the early steps of the generation process may, in hindsight, turn out to be \"errors\", and empowering the model with a mean to fix these errors could prevent their accumulation and improve performances. Iterative Denoising (ID) is introduced to this end, and further improvements are sought with Critical Iterative Denoising (CID).\n\nGiven a trained Discrete Diffusion Model (DDM), one may get an ID without any training: it is purely a matter of perspective and inference-time algorithm. Both DDM and ID use the current state $Z_t$ to predict a candidate denoised data point $z_1$. Whereas a DDM would obtain the next state $z_s$ (with $1<s<t$) by \"interpolating\" between $z_1$ and $Z_t$, ID applies the same forward (noising) process to $z_1$ as it would during training. If \"errors\" are made early on, there is some possibility for such a forward (noising) step to undo them.\n\nCID is more involved. The first step is to augment the system with a Bernoulli random variable $a_t$ that indicates whether each element has been corrupted (0) or not (1). The backward distribution has no dependencies in entries of $z_t$ for which $a_t=0$. The idea is to train a Critic model to predict $a_t$, then only update the entries that are predicted to be masked. An expression for what and optimal Critic should return is derived, and a GNN is trained on top of the predicted data to play that role.\n\nEvaluation is performed on molecular graphs and generic graphs. Despite both sharing the same denoiser, CID is not always superior to ID, but does provide improvements in some contexts, notably in the low Number of Function Evaluation (NFE) limit.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for your review, your careful reading and questions. We believe they will contribute to significantly improve our submission.\n\n## Answer to the questions\n\n### Question 1: \n\nWe broadly agree with your  summary in **Claim and Evidence** and specifically with your assessment in Claims 1, 3, and 4, and thank you for the thoughtful summary, which will help improve the clarity of our manuscript. As suggested, we will include a clear summary of the claims in the introduction.\n\n**Claim 1**\nWe will explicitly discuss the novelty of our contributions and include a reference to Bansal et al. (2023). We are grateful to the reviewer for pointing out this relevant work.\n\n**Claim 3** (and second part of Claim 1)\nRegarding the relationship between our Critic and the \"Planner\" introduced in Liu et al. (2025), we will add a concise discussion clarifying the distinction. We also acknowledge the emergence of concurrent works after our submission and will cite them accordingly.\n\n**Claim 4**\nAs correctly noted, Marginal ID captures aspects that Mask ID does not. While Mask CID tends to outperform Mask ID, it does not consistently outperform Marginal ID. Mask CID appears particularly effective in the low NFE regime. We agree that the choice between Marginal ID and Mask CID depends on the application and objective, and we will elaborate on this point in the revised discussion.\n\n\n### Question 2:\nThe error ranges are provided in the appendix. Considering these, the observed 1% difference in V.U.N. between Mask ID and Mask CID for Planar is not statistically significant, as both error margins exceed 5%.\n\nHowever, on another metric—spectral MMD—the difference is more substantial, clearly favoring CID. This suggests that the Critic captures structure not reflected in validity scores but visible in graph spectra.\n\nWe will clarify in the revised version that the V.U.N. difference is not significant, referring explicitly to the error bounds in the appendix. We will also note that the improvement offered by Mask CID in this case primarily appears in the spectral-based metric rather than in the validity.\n\n---\n\n## Other Comments Or Suggestions\nWe thank the reviewer for the detailed proofreading. We will implement the suggested corrections in the camera-ready version. Below are specific revisions:\n\n- Line 103: The notation was indeed confusing. We will revise the sentence as:\n    “We sometimes represent the univariate categorical distribution over the variable \\(p(z)\\) as a vector \\(\\vz\\), where the $i$-th component $z_i$ denotes the probability that $z$ belongs to the category indexed by $i$.”\n\n- Line 214: We will rephrase as:\n    “Our noising procedure is not a diffusion process in the traditional sense, as the state at time $t$ depends only on the initial state and not on the full trajectory or past states.”\n \n---\n\n## Essential References Not Discussed\n\nWe thanks the reviewer for these references. We note (for potential readers) that the 'concurrent works' were not publicly available at the time of submission. We will however include and discuss them in the camera-ready version.\n\n---\n\n## On the Relationship Between Our Critic and the Planner in [2]\nWe thank the reviewer, as well as reviewer *VwCn*,  who highlighted this connection. The revised version will include the following clarification.\n\nOur Critic and the Planner proposed in Liu et al. (2025) differ ins some fundamental aspects. \nThe Planner operates within the DDM setting, while our Critic is specifically designed for our Iterative Denoising (ID) framework.\nThe Planner determines, at each step $t$, which elements should be denoised (e.g., unmask), aiming to optimize the denoising order. \nImportantly, once an element is unmasked, it cannot be remasked in this framework.\nIn contrast, our Critic operates on the fully denoised instance and identifies which elements should be renoised. This allows for previously unmasked elements to be remasked. \nMoreover, as highlighted by reviewer whDB, our Critic comes with strong theoretical grounding. \nIn short, the Planner seeks to prevent compounding denoising errors via better ordering. Our Critic actively corrects such errors post hoc by leveraging renoising and multiple denoising iterations.\n\n---\n\n[1] Bansal et al., Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise. NeurIPS 2023. https://proceedings.neurips.cc/paper_files/paper/2023/hash/80fe51a7d8d0c73ff7439c2a2554ed53-Abstract-Conference.html\n\n[2] Liu et al. Think while You Generate: Discrete Diffusion with Planned Denoising. ICLR 2025. https://openreview.net/pdf?id=MJNywBdSDy\n\n[3] Kim et al. Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions. https://arxiv.org/pdf/2502.06768\n\n[4] Peng et al., Path Planning for Masked Diffusion Model Sampling. https://arxiv.org/abs/2502.03540\n\n[5] Nie et al. Large Language Diffusion Models. https://arxiv.org/pdf/2502.09992", "claims_and_evidence": "The manuscript does not explicitly give a summary of claims, so the following are my own understanding.\n\n### Claim 1: ID is novel\n\nID is so simple that my first thought was \"someone must have done it already\". However, all I've managed to find is how Cold Diffusion [(Bansal et al., 2023)](https://proceedings.neurips.cc/paper_files/paper/2023/hash/80fe51a7d8d0c73ff7439c2a2554ed53-Abstract-Conference.html) presents \"Naive Sampling\" in their Algorithm 1. I'm not aware of anyone presenting ID as a \"good thing\" in the past, and it may be that ID only makes sense in specific settings such as mask-based discrete diffusion contexts. As long as this situation gets clarified in an eventual camera ready, I'm ok with this aspect of the novelty question.\n\nThen comes the novelty of \"fixing\" the noise accumulation problem with a \"remasking\" approach. There are many concurrent works attempting this, including Kim et al. (2025), Nie et al. (2025) and Peng et al. (2025). All these approaches are closer in spirit to CID, as they involve a planner. Also, ID has the advantage to being very simple and theoretically well-grounded/justified (unlike some of the concurrent work). Again, I believe that this is novel, but warrants some clarifications/rephrasings/citations in the camera ready version.\n\n### Claim 2: ID addresses the error-accumulation issue\n\nI believe that this claim is supported both theoretically and practically.\n\n\n### Claim 3: CID is novel\n\nCID itself is almost certainly novel. The \"critic\" in CID can be related to a \"planner\" as defined in Liu et al. (2025), and see also the aforementioned concurrent work in Claim 1. Again, I believe that adding some clarifications (and citations) as to the extent of the novelty to the camera ready should suffice.\n\n\n### Claim 4: CID confers improvements over ID\n\nOverall, Mask CID beats Mask ID. However, for some task/metric combinations, Marginal ID (and other baselines) can capture something that that Mask CID cannot. As defined, CID may only be used with Mask noise, and this restriction may thus make it worse than ID. This is not a deal breaker, but I believe that it should be stated explicitly in the camera ready version.", "methods_and_evaluation_criteria": "The proposed methods and evaluations are standard and make sense. Some OBGN would have been nice.", "theoretical_claims": "I understand and sanity-checked the equations of the main text. I quickly browsed the appendix, but did not delve deeply in the proofs.", "experimental_designs_or_analyses": "Metrics are reported without error ranges. This is sadly common in the field.", "relation_to_broader_scientific_literature": "It is a paper of its time. Mask-based discrete diffusion, no conditioning on time, training a planner for remasking... As mentioned in Claims 1 and 3, there are many concurrent works aiming at the same general goal.", "essential_references_not_discussed": "Published work:\n\n- Bansal et al., Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise. NeurIPS 2023. https://proceedings.neurips.cc/paper_files/paper/2023/hash/80fe51a7d8d0c73ff7439c2a2554ed53-Abstract-Conference.html\n- Liu et al. Think while You Generate: Discrete Diffusion with Planned Denoising. ICLR 2025. https://openreview.net/pdf?id=MJNywBdSDy\n\nConcurrent work:\n\n- Kim et al. Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions. https://arxiv.org/pdf/2502.06768\n- Peng et al., Path Planning for Masked Diffusion Model Sampling. https://arxiv.org/abs/2502.03540\n- Nie et al. Large Language Diffusion Models. https://arxiv.org/pdf/2502.09992", "other_strengths_and_weaknesses": "I like the simplicity of ID, and the mathematical grounding of both ID and CID.\n\nI have reservations as to how performances on this type of benchmark would translate to concrete use-case scenario, but this is an issue pervading the whole generative graph modeling subfield: I don't think that I can fault the manuscript for this.", "other_comments_or_suggestions": "Please consider adding a clear summary of the claims in the introduction: readers (and reviewers!) like those.\n\nLine 103, left column: I'm not clear on the $Z=i$ notation here.\n\nLine 214, left column:\n> our noising procedure is not a diffusion process\n\nThis may raise the hairs of some readers. I propose adding some caveat such as \"(at least not in the standard sense)\". In a world with \"X is secretly Y\" papers, and things as weird as Cold Diffusion being considered diffusion...\n\nLine 253, right column: \"noising noising\"\n\nAt the start of Section 4.3, I suggest the addition of a complimentary reminder that $p_{\\alpha}(a)$ is the Bernoulli distribution defined above Equation (11). It took me a while...\n\nLike many recent models, ID and CID do not require an explicit time/noise dependency, which is a highly desirable feature. I initially thought that this was the \"time dependencies\" being discussed in the manuscript, but I then realized that it may only speak of the accumulation of errors. I do think that ID and CID have this desirable characteristic, and that it should clearly be highlighted in the manuscript. (It is not novel in itself though.)", "questions_for_authors": "My current score of 4 assumes satisfying answers to the following questions.\n\n### Question 1\n\nDo the authors agree with my general assessment in Claims 1, 3 and 4 above? If yes, do they consent to such edits in the camera ready? If not, please clarify the situation, and propose changes (if any) in the camera ready.\n\n### Question 2\n\nIs the difference between the V.U.N. of Mask ID and Mask CID for `Planar` significant? Mask CID is one point below, and it *shouldn't*. While a \"better\" model could potentially be \"worse\" on the \"unique\" and \"novel\" metrics, the manuscript states that all samples are unique and novel, so here V.U.N. actually amounts to the validity rate. Please discuss, and state what changes (if any) you would make in the camera ready.\n\n\n## April 14 update\n\nNow that my two fellow reviewers have replied, I revert my score to it's \"true\" value of 4.", "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": "PQx81d9bWt", "reviewer": "Reviewer_4gxJ", "summary": "This paper introduces Iterative Denoising, a novel framework to improve discrete diffusion and flow matching models for graph generation. Traditional discrete diffusion models suffer from error accumulation and propagation due to time dependencies in the noising process, particularly in mask diffusion. The proposed framework circumvents this issue by assuming conditional independence across time, effectively simplifying the diffusion process.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for the review. \n\n## Claim and Evidence\n\n**Paragraph 1**\n\nThe justification for our work is as follows:\n\n- We identify a limitation of discrete diffusion models—namely, the issue of compounding denoising errors.\n\n- We propose a theoretically grounded method that directly addresses this issue.\n\n- We empirically demonstrate that our approach significantly improves over discrete diffusion in a specific domain: graph generation.\n\n- By introducing a novel perspective on discrete denoising models, our work contributes to a deeper understanding of this model family, and  thereby contributes to advancing research in discrete generation. \n\nWe agree that further investigation is needed to assess the applicability of our method to other domains. However, it is common practice to first introduce a method within a research area before exploring broader applications. For example, Discrete Flow Matching, initially introduced for protein co-design, and only later extended to language modeling, code, image, and graph generation, is now widely adopted across tasks involving discrete data.\n\n**Paragraph 2**\n\nOur claim that the proposed method significantly outperforms existing discrete diffusion baselines' is supported by rigorous empirical comparisons, over strong baselines and over ablated models using identical denoisers to ensure a fair and controlled evaluation.\n\nWhile our approach is closing the gap between discrete and continuous denoising methods, we acknowledge that it does not yet surpass the strongest continuous model, DruM. Nonetheless, discrete and continuous denoising remain complementary methods with competitive performance. We believe, there is an interest in continuing the development of both. \n\nMoreover, we argue that paper in scientific conferences should aim to advance science, rather that selling direct real-world applications, particularly in a subfield where, as noted by reviewer whDB, benchmark performance does not translates directly to concrete use-case scenario. \n\n---\n\n## Methods And Evaluation Criteria:\n\nWe respectfully disagree with the interpretation attributed to us by the reviewer.\n\nContrary to the reviewer's statement, we do **not** attribute compounding denoising errors (CDE) specifically to the absorbing-state (mask) kernel. Rather, we argue that this issue affects **all** commonly used noise kernels for categorical data, including uniform and marginal kernels.\n\nAs clearly stated in the manuscript (lines 110–113, right column):\n“Consequently, the CDE issue also affects other common noise distributions, such as the uniform and marginal distributions.”\nThe statement comes at the end of a paragraph justifying it. \n\nFour kernels have been proposed in [1], two of which (discretized Gaussian and token embedding distance) are not applicable here as they are tailored to ordinal or embedded data. In addition to the mask kernel, our experiments focus on the marginal kernel, which we prefer over the uniform kernel based on evidence from Digress [2] that uniform noising is a poor choice for graph data.\n\nTherefore, we disagree with the affirmation that alternative kernels could mitigate the issue. In fact, we show, both theoretically and empirically, that all standard kernels (including mask, uniform,  and marginal) are susceptible to compounding denoising errors, and that our method robustly addresses this in both cases.\n\nTo the question of what advantages our approach offers over other diffusion kernels, we refer the reviewer to:\n\n- Sections 2 and 3: outlining the theoretical benefits of our approach.\n- Tables 3 and 4: demonstrating consistent empirical improvements across both marginal and mask kernels.\n\n---\n\n## Experimental Designs Or Analyses (Baselines)\n\nWe acknowledge the importance of the referenced models and include them in our literature review. However, we view baselines primarily as reference points for evaluation, rather than an exhaustive survey.\n\nWe deliberately selected three representative baselines:\n\n- DruM: current state-of-the-art.\n- GDSS: as best standard continuous diffusion.\n- Digress: as best (equivariant) discrete diffusion.\n\nAll are equivariant, ensuring a fair comparison with our own model. We will clarify this choice more explicitly in the revised version.\n\nIt is also important to distinguish baselines from ablations. Unlike ablation studies, baseline comparisons do not control for multiple factors such as architecture, training time, or hyperparameter. As a result, differences in performance should be interpreted cautiously.\n\nBy contrast, our ablation studies—comparing discrete diffusion with our Iterative Denoising method under controlled conditions—provide more meaningful evidence of the proposed method’s effectiveness.\n\nWe fully agree with the reviewer that exploring applications beyond graph generation is important. We consider this an exciting direction for future work.\n\n\n[1] Austin et al. , NeurIPS 2021\n\n[3] Vignac et al., ICLR 2023", "claims_and_evidence": "The main selling point of the proposed method is unclear. While the authors suggest that their method is broadly applicable to discrete data, they only apply it to graph generation and compare it with graph generative models. If the focus is on graph generation, more background on other graph generative models should be included. Conversely, if the authors wish to emphasize the model's broad applicability, empirical analyses in other domains, such as natural languages, are necessary.\n\nThe authors claim that the proposed method significantly outperforms existing discrete diffusion baselines. Given that the current application is limited to graph generation, I expect the proposed method to at least match the performance of other graph generative models. However, as shown in Tables 1 and 2, the baseline DruM appears to outperform the proposed method. If this is the case, please clarify the advantages of the proposed model over DruM.", "methods_and_evaluation_criteria": "The manuscript is motivated by the issue of error accumulation in the diffusion process, which the authors attribute to mask diffusion or discrete diffusion models with absorbing kernels. Instead of the proposed technique, other diffusion kernels, as described in [1], could be employed to mitigate this issue. What advantages does the proposed technique offer over these alternative diffusion kernels?\n\n[1] Structured Denoising Diffusion Models in Discrete State-Spaces, NeurIPS 2021", "theoretical_claims": "The proofs seem solid.", "experimental_designs_or_analyses": "The authors should consider applying their method to other domains. In the context of graph generation, several important baselines are missing, including a discrete diffusion model EDGE[1] and some other graph generative models like GraphRNN[2], EDP-GNN[3], SPECTRE[4].\n\n[1] Efficient and Degree-Guided Graph Generation via Discrete Diffusion Modeling, ICML 2023\n\n[2] GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models, ICML 2018\n\n[3] Permutation Invariant Graph Generation via Score-Based Generative Modeling, AISTATS 2020\n\n[4] SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators, ICML 2022", "relation_to_broader_scientific_literature": "The application scenario for the proposed method is unclear. The authors should elaborate on this aspect.", "essential_references_not_discussed": "The manuscript lacks a comprehensive background on graph generative models. A thorough review of related works in graph generation is necessary, including those mentioned in the Experimental Designs or Analyses section.", "other_strengths_and_weaknesses": "See above.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: I'd first like to thank the reviewers: unlike most papers with high variance in the scores, there was a substantial amount of discussion here and good signal all around. The cross discussion was particularly helpful. However, I ended up needing to read through the paper carefully myself to resolve some very divergent judgements on the paper's camera-readiness. 

After some work on my end, I'm inclined to weakly accept the paper if there's room. The positives are the paper is very well-written. The method is sound and I see high impact here in general modeling of discrete data (even if the work here focuses on graphs, these lessons can be applied elsewhere). However, I'm also inclined to agree with VwCn that the paper doesn't exactly meet the bar in terms of clarity (which impacts the perception of soundness), support for some of the key ideas, and appropriately placing the work in the field.

I'm rather divided on this paper, as I think in its current state it clearly has value. But I also think that with a non-trivial amount of additional work the paper could be much much better and have potentially much higher impact. So I'll defer to the conference distribution (room given other more camera-ready papers) but I ask the authors in the case they get lucky there: if this paper makes the cut, please follow through on the complains from the negative reviews.
Decision: Accept (poster)