Learning Flexible Forward Trajectories for Masked Molecular Diffusion
Authors: Hyunjin Seo
Taewon Kim
Sihyun Yu
Sungsoo Ahn
Venue: ICLR 2026 Poster (2026)
Keywords: Molecule Generation
Masked Diffusion Models
Molecule Diffusion Models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Masked diffusion models (MDMs) have achieved notable progress in modeling discrete data, while their potential in molecular generation remains underexplored. In this work, we explore their potential and introduce the surprising result that naively applying standards MDMs to molecules leads to severe performance degradation. We trace this critical issue to a *state-clashing problem*-where the forward diffusion trajectories of distinct molecules collapse into a common state, resulting in a mixture of reconstruction targets that cannot be learned with a typical reverse diffusion with unimodal predictions. To mitigate this, we propose **M**asked **E**lement-wise **L**earnable **D**iffusion (**MELD**) that orchestrates per-element corruption trajectories to avoid collisions between different molecular graphs. This is realized through a parameterized noise scheduling network that learns distinct corruption rates for individual graph elements, *i.e.*, atoms and bonds. Across extensive experiments, **MELD** achieves 100\% chemical validity in unconditional generation on QM9 and ZINC250K datasets, while markedly improving distributional and property alignment over standard MDMs on both conditional and unconditioned generation.

Published as a conference paper at ICLR 2026
LEARNING FLEXIBLE FORWARD TRAJECTORIES FOR MASKED MOLECULAR DIFFUSION
Hyunjin Seo Taewon Kim Sihyun Yu', SungSoo Ahn'
Korea Advanced Institute of Science and Technology (KAIST)', Polymerize" (bella72, maxkim139 39, sihyun. yu, sungsoo. ahn) (ann)kkan @kaist ac. kr
ABSTRACT
Masked diffusion models (MDMs) have achieved notable progress in modeling dis- crete data, while their potential in molecular generation remains underexplored. In this work, we explore their potential and introduce the surprising result that naively applying standards MDMs molecules leads to severe performance degradation. We trace this critical issue to a state- clae-ccashing problem where the forward diffusion trajectories of distinct molecules collapse into common state, resulting in mix- ture of reconstruction targets that cannot be learned with a typical reverse diffusion with unimodal predictions. To mitigate this, we propose Masked Element Learnable Diffusion (MELD) that orchestrates per-element cortuption trajectories avoid collisions between different molecular graphs. This is realized through a parameterized noise scheduling network that learns distinct corruption rates for individual graph elements, i.e., atoms and bonds. Across extensive experiments, MELD achieves 100% chemical validity in unconditional generation on QM9 and ZINC250K datasets, while markedly improving distributional and property alignment over standard MDMs on both conditional and unconditioned generation. Project page: https:/hhllyoollhaao httpl/yooollyaa.igguu..... githhb.io//.LLDD io/LDD
INTRODUCTION
Molecular generation critical variety of real-world applications, such drug discovery (Si- monovsky & Komodakis, 2018) and material design (Jia et al., 2024; Yang et al., 2023). However, the task remains challenging due to the extremely large and complex nature of the chemical space (Du et al., 2024). With the remarkable recent progress in deep generative models (Kingma & &Welling, 2013; Rezende & Mohamed, 2015; Austin et al., 2021; Naveed et al., 2023), many approaches have attempted to tackle this problem by training neural network that learns molecular distributions from large molecular datasets, demonstrating a strong promise in accelerating molecule discovery (Jensen, 2019: Jin et al., 2018; 88; Shi al., 2020; Jo et al., 2022; Vignac et al., 2023; Yiming al., 2025).
In particular, recent works have focused on exploring generative models based on denoising diffusion flow-matching models, (Jo et al., 2022; Lee al., 2023; Vignac et al., 2023; Kong et al., 2023; Jo et al., 2024; Liu et al., 2024a), to learn a molecular distribution, inspired by their great success in other data domains with scalability (Ho et al., 2020; Song al., 2020; Austin et al., 2021: Nichol & Dhariwal, 2021; Ma et al., 2024; Kingma et al., 2021; Sahoo et al., 2024b; Wan et al., 2025). These models learn recover original molecules from corrupted versions through denoising process, where the corruption typically involves altering types of atoms and bonds (e.g., changing a carbon atom to nitrogen, or single bond to a double bond).
Meanwhile, researchers have explored masked diffusion models (MDMs; Austin et al. 2021; Chang al. 2022; Shi et al. 2024; Sahoo et al. 2024a). Unlike conventional diffusion models that typically design diffusion processes in continuous space, MDMs are specialized for discrete data by defining diffusion process more suitable in discrete space. Specifically, MDMs define the forward process as element masking and train the model to infill the masked element during the reverse process. Intriguingly, MDMs show great stability and scalability, being comparable or even better than previous generative models for discrete data, such as autoregressive language models (Ziegler & Rush, 2019; Hoogeboom et al., 2021) or high-resolution eext-to-image generation (Chang et al., 2023). Despite their success in other domains, applying MDMs molecular graphs is still underexplored.

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Published as a conference paper at ICLR 2026
Deament-agnosicc
(b) Element- spenentt--pocfffescceeullii in
Fixed Schedule
MELD
Denolser
Dunciser
Stade Clasking
po(gla.)
Pg
Polg 9t)
Figure 1: Comparison between (a) element-agnostic noise scheduling and (b) clement-specific noise scheduling. The former results an issue denoted as state-clashing, leading to generation of invalid molecules. MELD mitigates this with element-speed specifi noise schedule, effectively orchestrating the forward process to minimize state-clashings.
In this work, we focus on applying MDMs to molecular generation. Surprisingly, unlike other domains, a naive adaptation of existing MDMs molecular graphs results in significantly worse performance, often generating distributionally misaligned structures. We argue that this phenomenon stems from a state-clasiing problem: Molecular graphs with different properties and semantics easily collapse into common intermediate state in the forward process (see Figure I(a) for an illustration). We attribute this to the usage of fixed noise schedules; element-agnostic masking rates across all nodes and edges. This hinders the learning process of the unimodal denoiser which predicts nodes and edges independently, by creating multimodal targets for reconstruction. Such mismatch forces the model to spread its probability mass into an averaged state creating samples that deviate from the true target distribution and, in some cases, violate chemical rules.
To address this, introduce MELD (Masked Element- molecular graph generation. The main idea of our method is to alleviate the state-clashing problem by proposing element-wise learnable forward process, which generates corruption trajectories in the way of minimizing the occurrence of potential collision. To this end, we introduce parameterized noise scheduling network to yield distinct corruption rates for individual graph elements (i.e., for nodes or edges). During training, we jointly optimize the forward (i.e., noise scheduling network) and the reverse process (i.e., MDM denoiser network). Intuitively, by assigning per-element trajectories, MELD organizes the forward process such that the probability of molecules being collapsed to the same intermediate state (see Figure (b)) is minimized). Through such evasion, MELD effectively learns to produce samples capturing the target molecular distribution.
wise Learnable Diffusion), a novel MDM for
We evaluate MELD on diverse molecular datasets, including QM9 (Ramakrishnan et al., 2014), Polymers (Thornton et al., 2012), ZINC250K (Irwin et al., 2012), Guacamol (Brown al., 2019), and synthetic graph benchmark (Martinkus et al., 2022). First, we demonstrate that MELD yields substantial improvements in distributional similarity over standard MDMs, while maintaining 100% validity. In conditional generation, MELD further enhances property alignment by up to 13.4% over state-of-the-att baseline. Finally, we show the scalability and generalizability of MELD in large-scale molecules and non non-molecule graph datasets.
Our contributions are threefold:
We identify a key limitation in applying standard masked diffusion models to molecular generation, the use of an element-agnostic noise schedule, which leads to frequent state-clashing.
We present MELD, a novel masked diffusion framework that mitigates the state-clashing problem by learning per-element noise schedules, allowing adaptive corruption trajectories tailored to individual molecular components.
MELD substantially improves the overall quality of generated molecules over standard MDM baselines, and surpasses existing molecular diffusion models in both unconditional and property- conditioned generation tasks. Moreover, its efficacy generalizes consistently large-scale molecule and synthetic graph benchmarks.
2
RELATED WORK
Masked diffusion models (MDMs). MDMs have emerged as a powerful generative modeling scheme for discrete data generation. Initially, D3PM (Austin et al., 2021) introduces absorbing
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mask token into the forward process and establishes a conceptual bridge between discrete diffusion and masked language modeling. Additionally, in image generation, MaskGIT (Chang et al., 2022) shows that generative modeling based unmasking enables fast and qualitatively comparable high- fidelity image synthesis compared with left-to-right autoregressive decoding. More recent efforts have further refined MDMMs to close the performance gap with autoregressive models (AR; Vaswani et al. 2017; Ziegler& & Rush 2019; Hoogeboom et al. 2021). Notably, MD4 (Shi et al., 2024) and MDLM (Sahoo et al., 2024a) show that the diffusion objective can be simplified as a weighted integral of cross-entropy and that the model can achieve state-of -fr results over prior diffusion models.
However, naive adoption of the MDM framework in molecular graph generation introduces unique challenges, termed as state-clashing problem. As molecular graphs exhibit higher symmetries while utilizing smaller vocabulary, the forward process of MDMs easily collapse distinct graphs into same intermediate state, hindering the learning process, as evidenced in Tables 1 and 2. We formulate this problem further in Section 4.1.
Diffusion models for molecules. The success of diffusion models for image (Rombach et al., 2022) and text generation (Li et al., 2022) has inspired researchers to explore diffusion models in molecule domain. A surge of studies (Vignac et al., 2023; Jo et al., 2022: Xie et al.. 2021; Kong al., 2023; Jo et al., 2024; Liu et al., 2024a) have been proposed to generate de novo molecules, competing with sequential models (Segler al., 2018; Jin et al., 2018; Shi et al., 2020; Jang et al., 2024a;b) that iteratively constructs a graph by adding graph elements. These efforts can be categorized into two approaches: (1) Score-based molecule diffusion approaches (Jo et al., 2022; Lee et al., 2023; Jo et al., 2024) adopt continuous noise on molecular graphs using stochastic differential equations (SDEs) (Song et al., 2020). They train score function to approximate reverse SDEs, relaxing discrete atoms/bonds into a continuous space. (2) Discrete diffusion seffusiio-bbsed approaches (Vignac et al., 2023; Liu et al., 2024a; Hua et al., 2024; Kerby& & Moon, 2024) apply discrete noise through Markovian transitions to nodes and edges in molecular graphs. Then they train a denoising neural network to reconstruct perturbed atom and bond types.
Despite progress in these two directions, masked diffusion frameworks remain underexplored for molecular generation. A preliminary application was explored in Kong et al. (2023), but it generates atoms in an autoregressive manner, limiting its ability to exploit the parallelized reconstruction of MDMs. In contrast, we propose MDMs for molecular graphs by focusing on the state-clashing problem occurring in the forward process, while preserving the parallelism inherent to MDMs.
3 PRELIMINARIES
We provide a brief overview of masked diffusion models for molecular generation. The goal is to generate molecular graphs g (z, e) from a data distribution (g)) where æ (z*) )== and (e = represent one-hot encoded node and edge features, each augmented with absorbing [mask] token. Following standard diffusion frameworks (Ho et al., 2020; Ma et al., 2024; Nichol & Dhariwal, 2021; Peebles & Xie, 2023), we consider a forward process go (g 9t-1) and a reverse process pa (g - Ig:), parameterized by é and respectively.
The forward process is defined as follows, where " denotes the marginal masking probabilities parameterized by di
if x [mask] if x ob
e ij [mask] if
(I)
if 1j = :
Tep
Most existing molecular diffusion models (Vignac al., 2023; Jo al., 2022; Lee et al., 2023; Jo et al., 2024; Liu et al., 2024a) have defined the corruption probability using a fixed, element-agnosicc noise scheduling function (i.e., 7:).
The denoiser predicts the original graph go by independently predicting nodes and edges. It trained to recover the original graph directly without recursive sampling (Vignac et al., 2023; Liu al.,
2024a), by minimizing the following loss objective:
C(0,0 6) = Br...g
Σ ISISN
+
Σ
log (e (e>)))))
(2)
1-
7
ISKJJN
7
3

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(a) Input Molecule
(b) Element- ognastic Schedule (power-law)
(c) MELD (Ours)
1.0
0.8
66666666
66666600
Normalized Entropy
arsccecce CCCNSeCN
CCCNCCCR
arcccccce CECNCCCK
aFccgecce CCENCCCA
0.6
0.4
+c00e066
86008600
86608006
arocccece
aFFecsees
0.2
2- Co flardsdunne
G00620610
606666640
666666640
Coloociko
0.0
Figure 2: Visualization prediction entropy for various molecule types. The first and second rows show prediction matrices with nitrogen bonds masked, while the third shows generations with chlorine bond masked. From left to right: (a) 3D renderings of the input molecules, (b) predictions from MDMs using fixed power law noise schedule, and (c) predictions from MELD. Brighter colors indicate lower uncertainty (i.e., higher confidence). The dark diagonal entries reflect enforced uniform predictions, as self-connections in molecules are not meaningful and are excluded from valid outputs. Note that predictions are being made for all locations, regardless of their entropy values.
Here, tt. denotes the derivative of Y,, with respect to I, while À balances node- and edge-level reconstruction, following prior work (Vignac et al., 2023; Liu et al., 2024a).
4 MELD: MASKED ELEMENT-WIST LEARNABLE DIFFUSION
In this section, we introduce MELD, a masked diffusion model (MDM) for molecular graph genera- tion that jointly learns per per-grap--elemen corruption rate and the denoising model. As we will explain. our proposed design alleviates the state-clashing problem (Section 4.1) by producing distinguishable forward trajectories for each molecular component (Section 4.2).
4.1 FORMALIZING THE STATE-CLASHING PROBLEM
In this section, describe the state-clashing problem which naturally arise for training MDMs graphs without learning the forward process, i.e., set yi.s to some constant for all node and edges (i,z). . In a nutshell, state-clashing refers to the phenomenon where semantically distinct molecules corrupted into the same intermediate state, due to the nature of the constant forward process in MDMs. Consequently, the model trained with such constant forward process struggles to infer the correct reconstruction target, resulting in outputs that fail to preserve structural or molccular coherence with target distribution (see Figure I for an illustration). This is particularly pronounced in molecules with symmetric motifs, to which the number of immediate parent states grows by the number of permutations that leave the motif invariant.
Formally, note that the diffusion model loss in Equation 2 can be expressed as:
E, (LLLpgglleee (g). po g:))], p(gig)) p(grlg)p(g).
(3)
The main problem is that p(g g.)) can be highly multimodal, i.c., there exists many graph with non-zero probabilities of p( (g). However, the parameterized diffusion model po(gig.) = HISENN Pe(x'(g) (g:) HI<vsYN (e gi) is unimodal, as it predicts each node and edge indepen- dently, typically resulting in single mode centered around an average graph. Furthermore, due to the mode-covering property of KL divergence, the reverse diffusion model trained with Equation tends to converge to high-entropy distribution-thh -th model compensates for its inability represent multiple modes by spreading its probability mass broadly around the single mode.
This is illustrated in Figure 2, where visualize the denoiser's prediction entropy when reconstruct- ing masked bonds in the given molecules. In the first two rows, we mask all catrogen-carbor bonds in and n--phenyleediamine. As masking removes the distinguishing nitrogen atoms, both molecules
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collapse into the identical symmetric benzene backbone, creating severe state-clashing scenario. Under an Semeent-gnostic schedule, the denoiser exhibits higher uncertainty when predicting the masked bond types, many distinct underlying configurations are compatible with the same cor- rupted state. Additionally, we visualize the denoiser prediction for Cchloro--ffuooroolleee when only the chlorine bond is masked. Due to the inherent asymmetry of the masked molecule, the state- clashing issue is less pertinent than Phenylenediamine isomers. Consequently, the denoiser shows increased prediction confidence even with -aement-agnostcc schedules, underscoring the necessity of addressing the state-clashing.
We note that this issue is not unique to MDMs, but it does become significantly more severe in their case. Masking operations tend to absorb diverse input graphs into indistinguishable intermediate states, whereas the probability of state-clashing in substitution-based discrete diffusion is orders of magnitude lower with realistic vocabulary sizes (see Appendix ?? for detailed qualitative analysis). Moreover, the effect particularly pronounced molecular graphs, which often contain structural symmetries and limited set of element types compared to other discrete domains.
4.2 MAIN ALGORITHM
Learnable element-wise embedding. To reduce state-clashing in forward diffusion trajectories across graph states, one should use information that distinguishes individual graph elements, which guides the noise scheduling network. One can consider incorporating graph positional encod- ings (Dwivedi et al., 2022; Ma et al., 2023) for conditioning. However, such encodings often fail to disambiguate elements when given symmetric structures such as those found in aromatic rings (Lawrence et al., 2025; Morris et al., 2024). Moreover, conditioning the noisy graph input in the noise schedule breaks the tractable closed-form marginal g(g: go) since the transition kernel becomes dependent on the current corrupted state, which eliminates the efficiency.
Thus, we consider learnable element-wise embeddings over the graph elements that assigns distinct masking rate, and use it for an input the noise scheduling network. Specifically, we assign learnable embedding matrix H E DxNN and consider its i-th column /t' as node-wise embedding of i-th node x', where N > 0 a number of nodes and D is the embedding dimension. For an edge {i E E, we set its embedding by hdd = h2' + h². In addition, we randomly permute columns of H during training to differentiate graph states that have the same numbers of nodes and edges.
Time-dependent noise schedule. We parameterize the noise scheduling network for each element (e.g., node) using power-law function, commonly used in Shi et al. (2024; 2025). Leveraging i-th node embedding h' as an example, our noise schedule Yi. is defined as:
%0 = - (1 - e) C") w (h'),,
where d&r denotes the softplus function. The same computation applies analogously to other nodes and edges. Consistent with Shi et al. (2024; 2025), we introduce a bounding constant e for numerical stability and fix 0.0001 in all experiments. Throughout this process, MELD naturally introduces element-specific masking rates, mitigating the collapse between distinct molecules that would otherwise persist under element-agnostic noise scheduling.
Maintaining gradient flow in discrete sampling. In discrete-space molecular diffusion frame- works (Vignac et al., 2023; Liu et al., 2024a; Kerby & Moon, 2024), the noisy graph at each timestep is obtained by sampling a single graph from a categorical distribution over nodes and edges (Equa- tion 1), computing the full expectation over g. ~ q(- (g) is intractable. However, such discretization introduces a discontinuity in the computational graph when parameterizing the forward process, impeding a gradient flow towards the learnable schedule parameters Φ. Thus, we adopt the Straight- (STGS) estimator (Jang et al., 2017), which provides a differentiable surrogate for discrete sampling. This formulation ensures the forward pass to utilize one-hot vectors for graph constructions, while the backward pass approximates them as continuous variables to enable end-to-end training.
Through Gumbel-Softmm
Let E R denote the logits, and 0 be the temperature parameter. We first compute a soft approximation of the categorical distribution Punn E [0. 1 via the Gumbel-Softmax:
exp((*k + gx)/n)
P Σn exp((= + 9)/n)
(5)
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Table Unconditional generation of 10K molecules on QM9 and ZINC250K datasets. The best and second best performances are represented by bold and underline.
QM9
ZINC250K
Method
Valid.
PCDL
NSPDK.
Scaf.1
Uniq.
Novel.
Valid.1
FCDJ
NSPDK]
Scaf.1
Uniq.1
Novel.1
Flow-based
MoFlow
100,00
GraphAP
74.43
5.63
0.021
0.305
88.64
86.59
68.47
16.02
0.044
0.067
98.64
99.99
GraphiDt
95.88
10.95
0.064
0.098
98.58
98.54
90.61
33.55
0.177
0.000
99.63
100,00
Continsons diffissin
EDP-GNN
47.52
2,68
0.005
0.327
99.25
86.58
82.97
16.74
0.049
0.000
99.79
100.00
GDSS
100,00
GruM
99.69
0.11
0.0002
0.945
96.90
24.15
98.65
2.26
0.0015
0.530
99.97
99. 98
Discrete diffusion
DiGiress
98.19
0.10
0.0003
0.936
96.67
25.58
94,99
3.48
0.0021
0.416
99:97
99 99
Masked diffiszion
GraphARM
90.25
1.22
0.002
NIA
95.62
70.39
88.23
16.26
0.055
N/A
99.46
100.00
MDM w/ cosine
100,00
3.67
0.009
0.653
85,96
69.85
100.00
25.41
0.051
0.001
99.99
100.00
polynomial
0.010
0.053
MDM w/ power-law
100.00
3.62
0.007
0.628
91.30
76.66
100.00
26.09
0.068
0.001
100.00
100.00
MELD (Ours)
100.00
0.09
0,0002
0.947
33.55
100,00
1.51
0,0006
0.559
100.00
99.96
where 9& log( log( (xx) is gumbel noise with uy Unif(0, 1] and 2₁ is the k--th element of
the logits 2. A discrete one-hot vector Phard E {0, 1 is then obtained by taking the index with the highest probability:
{1 (0
if = K-*
k- arg max Pson &
Phard,
(6)
otherwise
To retain gradient flow, the straight-through estimator to combine the discrete and continuous components, i.e., set Phard sg(Pson) + Psoft. where sg(.) denotes the stop-gree gradient operator. This ensures that the forward pass uses the discretized one-hot vector p = Pbadd while the backward Op... pass treats p as the continuous Psan. allowing gradients propagate through 2, i.e., az 8a
Permutation equivariance. While strict equivariance is essential for molecular representation learning, e.g., mapping isomorphic graphs to identical embeddings, graph generation only requires the learned distribution be permutation invariant. Our method achieves this invariance not by constraining the architecture, but by marginalizing over permutations. Formally, we model the probability of a graph g as the expectation over all possible node orderings i.e., p(g) Σn p(g, T), where the summation is over the possible permutation л. Our randomized permutation strategy corresponds to maximizing the evidence power-bound (ELBO) of this marginal log- liggliiklihooo::
logr(g) log r(gg 2
E, Ez x)] + const.
This stochastic symmetrization is well-established paradigm in graph generation, widely used in autoregressive models (You et al., 2018b; Kong al., 2023), which are inherently node-order dependent yet generate valid, invariant graph distributions.
Domain specialization and applicability. principle, MELD is applicable to non-molecular data. However, we note that other discrete data such as text or protein sequences typically involve larger vocabularies and fewer structural symmetries. Consequently, the risk of collapsing distinct inputs into identical intermediate states is lower, and the relative benefits of MELD may be less pronounced in such settings. Nevertheless, to show the generality of our approach, we include additional experiments general graph with constrained number of nodes and edges in Section 5.4.
EXPERIMENTS
5.1 EXPERIMENTAL SETUP
We evaluate MELD on unconditional and property-conditioned molecular generation tasks. For unconditional generation, in line with prior work (Jo et al., 2024; Kong et al., 2023; Jo et al., 2022), we use QM9 (Ramakrishnan et al., 2014), ZINC250k (Irwin et al., 2012), and Guacamol (Brown al., 2019) datasets. For conditional generation, we adopt the Polymer dataset (Thornton al.,

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Table 2: Property-conditione generation of 10K Polymers on three gas permeability properties and synthetic score. The numbers in parentheses in Valid. represent the validity without correction. The best and second best performances are represented by bold and underline.
General Quality
Property Alignment
Method
Valid.
Cover.
Divers,
Frag.t
FCD.
Syuth.]
Og Perm.
N₂ Perm.l
CO2 Perm.
MAE]
Molecude Optimization
GraphGA
100.00 (N/A)
9.19
1.3307
1,9400
1,9489
1.888
MARS
100.00 (NIA)
11/11
83.75
0.928
7:56
1.1658
1,5761
1.8327
1.6074
1,546
LSTM-HC
99.10 (N/A)
10/11
89.18
0.794
18.16
1.4251
1,1003
1.2366
1,0772
1,210
JBVAE.BO
100.00 (N/A)
10111
73.66
0.729
23.59
1,0714
1,0781
1.2352
1,0978
1,121
Continauus difftarion
GDSS
92.05 (90.76)
W11
75.10
0.000
34.26
1.3701
1.0271
1.0820
1,0683
1.137
MOOD
98.66 (92.05)
11/11
83.49
0.023
39.40
1,4019
1.4961
1.7603
1,4748
1,533
Discrete diffassio
DiGress v2
98.12 (30.57)
11/11
91.05
0.278
21.73
2.7507
1.7130
2.0632
1.6648
2.048
GraphDiT
82.45 (84.37)
11/11
87.12
0.960
6.64
1.2973
0.7440
0.8857
0,7550
0.921
Masked Aiffasion
MDM w/ cosine
15.95 (37.16)
11/11
89.91
0.307
26.45
2.1745
1.5035
1.7755
1.4974
1.739
MDM w/ polynomial
18.61 (60.32)
11/11
88.44
0.237
29.32
2.0041
1,6805
1,9846
1,6468
1.829
MDM w/ power-law
17.31 (53.64)
11/11
89.08
0.401
26.56
2.0145
1.4100
1.6536
1.4030
1.620
MELD (Ours)
(96.5)))
0,974
5.93
1,1398
0.6433
0.7596
0.6496
0.798
2012) introduced in Liu et al. (2024a), which conditions homopolymers on three gas permeability constraints and synthesizability scores. We compare against recent baselines with standard metrics for both tasks, following established setups (Liu et al., 2024a; et al., 2022; 2024). See Section C for detailed description of each method and metrics. Our implementation employs the diffusion transformer (Peebles& Xie, 2023) as the denoising network within a masked diffusion framework. further apply classifier-free guidance (Ho & Salimans, 2021)
For property-cone condiiioned generation, as implemented in (Peebles & Xie, 2023; Liu et al., 2024a). Unless otherwise noted, all experiments use the same backbone across standard MDMs and MELD.
5.2 MAIN RESULTS
Unconditional Generation. We present the results of MELD on QM9 and ZINC250K datasets for unconditional generation. Remarkably, MELD substantially enhances distributional similarity while maintaining perfect validity, shown in Table 1. On the QM9 dataset, our method outperforms GraphARM (Kong et al., 2023), the autoregressive masked diffusion baseline, with up to 91 % reduction in FCD and NSPDK. Moreover, it significantly improves the NSPDK by up 98% from standard MDMs.
On the more challenging ZINC250K dataset, which includes larger molecules and richer atom types, MELD achieves taate-of-the-art results on out of 6 metrics, surpassing GruM (Jo et al., 2024), the strongest baseline. It also consistently improves over masked diffusion baselines key metrics including FCD, NSPDK, and scaffold similarity (Scaf.). In contrast, standard MDMs exhibit degenerate behavior, with FCD 91.4% higher and Scaf. 99.8% lower than the best diffusion-based baselines, suggesting that elenent-agnostic schedulers yield valid but distributionally misaligned molecules.
Property-conditione Generation. Next, we evaluate MELD on conditional generation using the Polymer dataset (Thornton et al., 2012), with results summarized in Table 2. Overall, MELD establishes new state-of-the--rt in property alignment, with 13.4% reduction in average MAE relative to GraphDiT (Liu et al., 2024a). Apart from GraphDiT, no existing method can satisfy multiple property constraints simultaneously: LSTM-HC achieves strong synthesizability MAE but fails under gas permeability targets. DiGress v2 (Vignac et al., 2023), despite leveraging classifier guidance (Dhariwal & Nichol, 2021), incurs substantially higher MAE across most conditions. Beyond alignment, MELD also improves generative quality. surpassing FCD and tragment-based similarity (Frag.) over the previous best. Consistent with earlier work (Ho & Salimans, 2021; Liu et al., 2024b), we observe an inherent trade-off between property alignment and sample diversity. Importantly, our method addresses the slateccaahhng issue prevalent in MDMs: whereas element- agnostic schedule results in generating low low-quality molecules. our learnable, element-wise noise schedule enhances validity by a factor of five and improves property alignment by an average of 50%.
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Table 3: Ablation study of MELD with varying noise scheduling approaches. Y without and % denote fixed and learnable schedules, respectively. V.U.N. denotes composite score for Validity, Uniqueness, and Novelty.
Schedule type
Method
FCD.,
NSPDKJ
Scaf.1
V.U.N.
Power-law
26.09
0.0683
0.001
100.00
Fixed y
DiffusionBERT (He et al., 2022)
1.95
0.0009
0.491
100.00
GenMD4 (Shi al., 2024)
3.19
0.0017
0.429
100.00
TabDiff (Shi et al., 2025)
2.15
0.0009
0.486
99.99
Learnable No
MELD (Ours; Node)
1.63
0.0009
0.536
99.99
MELD (Ours; Edge)
1.73
0.0009
0.525
99.99
MELD (Ours; Node + Edge)
1.51
0.0006
0.559
99.96
SAU & duat
Timestop
T-1
T/2 T/4
T/8
T/16
T/32
Figure 3: Comparison between fixed power-law scheduling and MELD during reconstruction. With the learnable noise schedule, MELD achieves faster recovery than standard MDMs.
5.3 ABLATION STUDY
We evaluate several learnable scheduling strategies on ZINC250K (Irwin et al., 2012), as summarized in Table The first row reports a standard MDM with power-law function, while the second-to- and third-to-last rows correspond to element wise learnable scheduling applied only to nodes or edges. Rows two through four present advanced scheduling strategies from prior work. DiffusionBERT (He et al., 2022) employs fixed spindle noise schedule decided by class-wise entropy; GenMD4 (Shi et al., 2024) is another class-wise scheduling variant where each atom and bond type has its own learned corruption rate; and TabDiff (Shi al., 2025) introduces single corruption rate shared across elements within the same column, analogous to node and edge-level schedules, e.g., all nodes sharing the same schedule. The final row corresponds the full element-wise schedule of MELD.
As depicted in the table, all alternative noise schedules fall short of optimal gains in key metrics such FCD and Scaf., an effect attribute to their limited ability of reducing state-clashing. For instance, employing GenMD4 noise scheduling can remain limited in resolving the state-clashing:
delaying the corruption all carbon atoms relative to nitrogen in o-Phenylencdiamine (Figure 2) may still result in symmetric benzene ring. In contrast, our full per-element corruption (MELD) delivers further reductions in distributional similarity metrics, demonstrating its fine-graie ine-grained control.
5.4 QUALITATIVE ANALYSIS
Reverse process of MELD. Figure compares MELD with standard MDM. Corrupted nodes and edges are shown as [mask] and dashed lines, respectively. Under unified noise scheduling, unmasking proceeds relatively slowly: time / = T/4 only HCI and two carbon atoms begin to emerge, ultimately leading to a poorly-designed molecule. In contrast, MELD reconstructs fragments earlier relative to elenent-agnostic schedule, where larger amount of atoms already reveal from step t = T/4. Similar phenomena can also be found with more examples in Section D.7.
Computational cost analysis practice, the computational and memory overhead introduced by MELD is negligible since it only adds learnable embedding matrix H to the existing transformer- based architectures. To validate this, we report (1) total number of parameters, (2) FLOPs, (3) peak memory usage, and (4) single-step runtime across various molecular sizes ([Vl [10, 50, 100, 200)
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Table Computational cost analysis with varying molecular sizes (batch size 1). All experiments were conducted on an NVIDIA GeForce RTX 4090 GPU and an AMD EPYC 7K62 48-Core Processor. Runtime values are averaged over random seeds.
#Atoms
Method
#Params (M)
FI.OPs (GMac)
Peak memory (MB)
Runtime (sec)
MDM
156.23
2.06
622.76
0.056 + 0.005
10
MELD
156.24
2.06
623.19
0.056 0.006
MDM
157.22
9.88
680.40
0.077 0.007
50
MELD
157.23
9.93
692.31
0.087 + 0.008
MDM
158.46
19.74
755.12
0.093 1 0.008
100
MELD
158.47
19.94
788.71
0.098 1 0.008
MDM
160.93
39.77
929.09
0.132 0.007
200
MELD
160.94
40.59
1067.64
0.165 + 0.008
for MELD and standard MDM in Table 4. Our results demonstrate that MELD introduces only about 0.01M additional parameters with negligible computational overhead, regardless of the input size.
Scalability to large molecules. We fur- ther evaluate MELD on the large-scale Guacamol dataset (Brown et al., 2019) fol- lowing the standard protocol used in prior work (Vignac et al., 2023). As demon- strated Table 5, MELD surpasses all diffusion-based baselines (Vignac al., 2023; Xu et al., 2024) while achieving 100% validity. Notably, this performance is obtained with 70% reduced training epochs (300 epochs) than DiGress (1000 epochs), emphasizing both efficiency and empirical gains.
Table 5: Performance comparison large-scale Gua- camol dataset. The metrics are transformed such that higher values indicate better performance.
Method
Valid.1
Novel.
KL div1
FCD1
ConGress (Vignac
DiGress (Vignac et al., 2023)
85.2
100.0
99.9
92.9
68.0
DiaCo (Xu et al., 2024)
96.6
100.0
99.9
92.6
59.7
MELD
100.0
100.0
100.0
93.4
68.8
Quantifying state-clashing problem. Here, we assess state-clashing phe- nomenon by measuring the number of distinct intermediate graph states at each timestep, shown in Table 6. Specif- ically, we sample molecules with a fixed graph size and employ graph isomorphism-basee based method (Cordella al., 2001) to count unique graphs.
Table 6: Number of unique graph states across varying timesteps in ZINC250K, averaged over 3 seeds.
Method
T-100
T-75
T:50
T-25
T-1
MDM cosine
131.0
122.3
63.0
14.7
1.7
MDM polynomial
131.0
131.0
131.0
103.0
13.3
MDM w/ power-law
131.0
131.0
131.0
126.0
8.7
MELD
131.0
131.0
131.0
131.0
17.3
higher count of unique graphs indicates fewer state-clashing. Due to the extreme cost of isomorphism algorithm, we sample 131 molecules with nodes from the ZINC250K dataset for the evaluation. The results show that MELD preserves greater structural diversity at later timesteps compared to any standard MDMs.
It is important to note that MELD is not intended to eliminate state-clashing entirely, but to reduce the chance its occurrence, particularly the early and intermediate timesteps. Inevitably, some clashes remain, e.g., all graphs converge fully masked state, but these unavoidable cases only affect a small portion of decisions near the prior distribution and therefore does not compromise its overall effectiveness.
Generalizability on synthetic graph. To assess generalizability of MELD on other discrete graph domains, we bench- mark MELD against two strong molecular diffusion models, DiGress and GruM, on SBM (Martinkus et al., 2022), synthetic graph benchmark. Following the standard evaluation protocol (Vignac et al., 2023;
Table 7: Performance comparison on synthetic graph domain (SBM).
Method
Degree
Cluster]
Orbit
Spectral
V.U.N.t
DiGress GruM MELD
0.0013
0.0498
0.0434
0.0400
74.00
0.0007
0.0492
0.0448
0.0050
85.00
0.0005
0.0506
0.0381
0.0047
97.50
Jo et al., 2024), compute the maximum
discrepancy (MMD) across four key graph statistics.

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Published as a conference paper at ICLR 2026
As reported in Table 7, MELD outperforms the baselines most metrics, with notable gains in V.U.N. (a composite score for validityumiqqeenes//ovelly and Orbit.
CONCLUSION
In this work, we investigated masked diffusion models (MDMs) for molecular graph generation and identified a central limitation, which term state-clashing To address this, we introduced MELD, a masked diffusion model that learns element-wiss forward trajectories through a parameterized noise scheduling. Extensive experiments show that MELD consistently outperforms standard MDMs and prior duffusion-based methods in both unconditional and property-conditione molecular generation.
ETHICS STATEMENT
From a broader perspective, MELD has a potential to accelerate molecular discovery and reduce the need for costly and time-intensive wet-lab experiments. thereby contributing to advancements in drug design and material science. However, as with any generative technology. there exists the risk of misuse, including the malicious design of toxic harmful compounds. We advocate for the responsible deployment of such models for the safe integration into real-world workflows.
REPRODUCIBILITY STATEMENT
We provide the source code and setup for our key experiments, with detailed configurations described in the appendix. The implementation has been carefully verified, and we empirically confirm the validity of the proposed method.
ACKNOWLEDGMENTS
This work was partly supported by Institute for Information & communications Technology Planning & (IIT)) grant funded by the Korea (MV)) (RS- RKS-2019-1119007 Artificial In- telligence Graduate School Support (rogram(KAIST)) National Research Foundation of Korea(NRF) grant funded by the Ministry of Science and ICT(MSIT) (No. Research Development Center) Cooperative Hub Program through the National Research Foundation of Korea(NRF) grant funded by the Ministry of Science and ICT(MSIT) (No. RS-2024-00436 Institute of Information & Communications Technology Planning & (valuation(IITP) grant funded by the Korea government(MSIT) (RS-2025 0025-02304967, AI Star Fellowship(KAIST) Supercomputing Center with supercomputing resources including technical support (KSC-2024-CRB) 0535). We also thank Polymerize for their valuable support in improving the manuscript.
84), GRDDCGGlobal
and the National
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A
MORE RELATED WORK
Molecule optimization. Optimization- methods generate molecules by iteratively refining can- didates assembled from predefined vocabulary of fragments, aiming to align with desired property constraints. These approaches typically employ techniques such as genetic algorithms (Jensen, 2019), 19), Bayesian optimization (Shahriari et al., 2015; Jin et al., 2018: Zhu al., 2023), and goal-directed generation (Mollaysa et al., 2020; You et al., 201 2018a). Representative examples include (Jin et al., 2020: 2018; Xie al., 2021; Fu al., 2022), which utilize predefined subgraph motifs or scaffolds to ensure chemical validity during the generation process. These methods rely on diverse strategies including Markov sampling to sparse Gaussian processes and optimize molecules based on property- specific scoring functions. particular, often adopts reinforcement learning, where generation policy is updated to maximize property-driven reward function. Despite their strengths, existing optimization-basse approaches remain limited in conditional generation settings. Specifically, they require full re-optimization for each new property configuration when tasked with generating molecules that precisely match target properties, rather than simply increasing or decreasing property values. This results in a high training complexity and limits their scalability (Aung et al., 2024; Xia et al., 2024).
generation (Mollaysa et al., 2020; You et al., 201 8a), in
Learnable noise scheduling. Several works have explored learnable corruption process to optimize the forward trajectories in images and text. In continuous-space diffusion models, Kingma et al. (2021) introduces learnable scalar noise schedule as function of time, enabling variance reduction in evidence lower bound (ELBO) estimation. Extending this, Sahoo et al. (2024b) proposes a multivariate, data-dependent noise schedule, showing that non-scalar, adaptive diffusion process can further tighten the ELBO by aligning the forward process more closely with the true posterior. In discrete masked diffusion, Shi et al. (2024) generalizes the corruption process to allow class- dependent masking rates across tokens, prioritizing semantically important tokens during generation. Shi et al. (2025) adopts reature-wise noise schedule for tabular data, where a single corruption rate is shared across elements within the same column. Additionally, Schrödinger bridges based approaches (Peluchetti, 2023; De Bortoli et al., 2021; Shi et al., 2023) formulate generative modeling as learning expressive, path-wise forward process by solving entropy regula-erggllrizzed optimal transport problems over path spaces.
It is noteworthy that the design philosophy of MELD is built upon the state-clashing, a critical issue that has not been addressed these work nor in the molecular diffusion literature (Vignac et al., 2023; Jo et al., 2022; 2024; Liu et al., 2024a; Lee al., 2023). While employing the learnable forward process, our work departs from existing methods by introducing graph clement-wise parameterization of the forward diffusion, specifically to avoid trajectory collisions between semantically distinct molecules. Moreover, we explicitly target and resolves the intermediate state degeneracy unique to discrete molecular graphs, while Schrödinger bridge-based approaches neither address structural collapse in discrete settings nor differentiate forward paths across individual graph elements.
B LIMITATIONS
While element-wise noise scheduling significantly mitigates the state-clashing issue, it may not fully address the inherent multimodality when a large portion of molecules are masked at later diffusion steps. This is especially pronounced later diffusion steps, where majority of the graph elements are masked, making it challenging to distinguish them. Nevertheless, these unavoidable cases only affects small portion of corruption near the prior distribution and therefore does not compromise the overall efficacy of our method.
C EXPERIMENTAL SETUP
Implementation details. We follow the evaluation protocols and dataset splits adopted in prior works: for unconditional generation, we adopt the setup from Jo et al. (2024), and for property- conditioned tasks, we follow the procedure outlined in Liu et al. (2024a). We provide the detailed
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Table Dataset statisties.
Dataset
#(Graphs)
#(Nodes)
#(Node types)
#(Edge types)
QM9
133,985
[V] <
4
3
ZINC250K
249.555
M S 38
9
3
Polymers 553
11
statistics of each dataset in Table 8. During training for unconditional generation, we apply an exponential moving average (EMA) to the model parameters, consistent with the training framework in Jo al. (2024). For conditional generation, we utilize the implementation strategies proposed in Peebles & Xie (2023); Liu et al. (2024a), including condition vector encoders and adaptive layer normalization (AdaLN). Across experiments, we use transformer-based denoising model (Peebles & Xie, 2023) with 6 layers, a hidden dimension of 1152, and 16 attention heads. The noise scheduling network is parameterized as a two-layered MLP with SiLU activation with hidden dimension set as 64. We train all models using the AdamW optimizer with no weight decay. We provide detailed training setups compared with representative baselines (Jo et al., 2024; Vignac et al., 2023; Liu al., 2024a) in Table 9. The FLOPS comparison with these baselines is shown in Table 10. Models are implemented in Py Paszke et al. (2019) with Py PyTorch Geometric Fey & Lenssen (2019). Experiments were conducted on machines equipped with NVIDIA RTX 3090 and 4090 GPUs (24 GB) and AMD EPYC 7543 32-Core CPUs (64 cores total). Note that MELD does not rely on increased GPU count specialized accelerators compared to baselines. For reference, the results of Jo et al. (2024) were obtained using RTX 3090 and 2080 Ti. while Liu et al. (2024a) used an A6000 GPU. These configurations all fall within similar class of commodity GPUs, and none of the compared methods (including MELD) benefits from substantially larger compute bugets or hardware advantages.
Table Training details comparison with representative baselines.
Datasets
Methods
Total epochs
Diffusion sampling step
Learning rate
Scheduler
Backbone
DiGress
1000
1000
Cosine
Graph transformer
QM9
GruM
1000
1000
2e
Linear
Graph transformer
MKLDD
1500
300
2m-4
Element wise
Diffusion transformer (DIT)
DiGress
500
1000
2e
Cosine
Graph transformer
ZINC250K
GruM
500
1000
2e-4
Linear
Graph transformer
MELD
600
500
2e 4
Element-wise
Diffusion transformer (DIT)
Polymers
GraphDiT
10000
500
20 4
Cosine
Diffusion transformer (DiT)
MELD
15000
200
2b-4
Element wise
Diffusion transfommer (DiT)
Baselines. We consider various recent baselines for conditional and unconditional generation; following experimental setups of prior works (Liu et al., 2024a; Jo et al., 2022; 2024).
Unconditional Generation: First, we consider three flow-based models as baselines: MoFlow (Zang & Wang, 2020), GraphAF (Shi et al., 2020) and GraphDF (Luo al., 2021), three continuous diffusion models: EDP-GNN (Niu et al., 2020), GDSS (Jo et al., 2022), and GruM (Jo et al., 2024), and one discrete-diffusion model: DiGress (Vignac al., 2023). Additionally, we compare MELD against GraphARM (Kong et al., 2023), a method that employs mask tokens as absorbing states but generates tokens (i.e., nodes) autoregressively.
Conditional Generation: We consider four optimization- based frameworks as baselines: GraphGA (Jensen, 2019), MARS (Xie et al., 2021). LSTM-HC (Neil et al., 2018), and JTVAE- BO (Jin et al., 2018), two continuous diffusion models: GDSS (Jo et al., 2022) and MOOD (Lee et al., 2023), and two discrete diffusion models: DiGress v2 (Vignac et al., 2023) integrated with classifier guidance and GraphDiT (Liu et al., 2024a).
Metrics. Following the evaluation protocol in previous work (Liu et al., 2024a: Jo et al., 2022; 2024), evaluate the performance of our framework using the following metries:
Unconditional Generation: We use 10,000 generated samples for evaluation using the following six metries: (1) Valid., the proportion of chemically valid molecules; (2) Frechet ChemNet Distance (FCD; Preuer et al. 2018), a distributional similarity score of ChemNet embeddings between generated and reference molecules; (3) NSPDK (Costa & De Grave, 2010), graph
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Table 10: Computational cost analysis comparing various methods (GruM, DÝT, DiGress, and MELD).
We report the average and standard deviation values of processing molecule size of [V = 100, with a batch size of 32, upon forward passes. All experiments were conducted on an NVIDIA GeForce RTX 4090 GPU and AMD EPYC 7K62 48-Core Processor.
Baselines
Masked Diffusion Models
Metric
GruM
DiT
DiGress
Fixed Poly.
Fixed Cosine
TabDiff'
MELD
FLOPss
209.9 0.1
314.7 +: 0.1
209.4 + 0.1
315.2 + 0.1
315.2 + 0.1
318.5 = 0.1
318.5 1 0.1
Exec (s)
1.12 + 0.u3
0.91
+ 0.01
1.29
= 0.04
0.94 # 0.02
0.94 + 0.02
0.95 = 0.02
1.16 + 0.03
Peak (GB) 19.16 + 0.05
2.74 + 0.01 19.15 0.06
2.74 + 0.01
2.74 + 0.01
3.27 = 0.02
2.77 + 0.11
kernel metric that quantifies topological similarity the reference set; (4) Scaf., scaffold-level similarity score; (5) Uniqueness, the proportion of valid molecules that are structurally distinct within the generated set; and (6) Novelty, the fraction of valid molecules not in the training data.
Conditional Generation: We generate 10,000 samples and assess their overall quality using the following criteria: (1) Valid., (2) Cover, the heavy atom type coverage: (3) Divers., the diversity similarity metric; and (5) FCD.
among the generated molecules; (4) Frag., a fragment- We also report Property Alignment, measured as the mean absolute error (MAE) between target properties and the corresponding oracle-evaluated scores of generated molecules.
To compute property alignment, we follow the setup of prior works (Liu et al., 2024a; Gao et al., 2022), employing a random forest model trained on molecular fingerprints as an oracle function.
We report official baseline results (except for Kong et al. (2023)) from Jo et al. (2024) for unconditional generation. For Kong et al. (2023), we generation and Liu et al. (2024a) for property-conditiones
take the results from the original paper. To evaluate the efficacy of our method in remedying state- clashing, we perform additional ablative studies against fixed-scheduling mechanisms often adopted in masked diffusion models; namely cosine (MDM w/ cosine), polynomial (MDM w/ polynomial), and power-law (MDM w/ power-law) scheduling functions. We train all vanilla MDM variants as well as MELD under identical training budgets. The corresponding standard deviation results reported in Table 11 and Table 12. Note that the standard deviations for GraphARM and the annditional-generatin baselines are not available.
D FURTHER EXPERIMENTS AND ANALYSIS
D.1 COMPARISON WITH MORE BASELINES
We present performance comparison between MELD and additional baselines Huang et al. (2023a:b). Note that the reproduced baseline performance differs from those in the original papers due to differences in formal charge assignment and bond order handling within the graph-to-molecule conversion pipeline, which we adopt from Jo al. (2022: 2024). As shown in Table 13, MELD performs better or similar to CDGS for most of the metrics except for Wasserstein distance for logP scores. The performance is comparable to JODO, with better scores for FCD, WD/logP, WD/QED, WD/SA, worse scores for NSPDK, Scaf, and similar scores for Valid. Uniq. and Novel. Importantly, MELD attains this performance with significantly fewer total training iterations (7.6x reduction compared to CDGS; 5.7x compared to JODO) and with half the diffusion sampling steps, underscoring the efficiency of our method.
D.2 ANALYSIS OF ELEMENT- WISE LEARNED EMBEDDING
Our design philosophy of learnable em- bedding in MELD is focused on reducing the chance of state-clashing problem by making each graph elements distinct and unique. As a result, our method can dis- tingussh graph elements even within the symmetric motifs, which is often difficult to be discriminated using existing graph positional encodings (Dwivedi et al., 2022; Ma et al., 2023).
Table 14: Average cosine similarity between pairs of h and h in benzene ring.
Cosine similarity Leamed H (MELD)
Random walk embedding
Nodes (f') Edges ( All
0.103
0,237
0.192
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Table 11: Unconditional generation performance MELD with standard deviation. The baseline results are taken from Jo et al. (2024).
QM9
Method
Valid.1
FCDJ
NSPDK]
Scaf.1
Uniq.1
Novel.1
Flow-bused
MoFlow
91.36 + 1.23
4.47
0.595
0.017
0.003
0.145 + 0.052
98.65 + 0.57
94.72 + 0.77
GraphAF
74.43
5.63 0.259 10.93
0.021
0.003
0.305 1: 0.056
88.64 d: 2.37
86.59 1,95
GraphDF
93.88 + 4.76
0.038
0.000 0.064
0.098 + 0.106
98.58 + 0.25
98.54 + 0.48
Continaous diffission
EDP-GNN
47,52 = 3.60
2.68 + 0.221
0.005 0.001
0.327 + 0.115
99.25 + 0.05
86.58 : 1.85
GDSS
95.72 1.94
2.90 0,282
0.003
0.000
0.698 +: 0.020
98.46 ::
86.27 2.29
GruM
99.69 + 0.19
0.11 0.006
0.0002 + 0.000
0.945 + 0.005
96,90 + 0.15
24.15 + 0.80
Discrete diffiasion:
DiGress
98.19 = 0.23
0.10 + 0.008
0.0003 + 0.000
0.936 + 0.003
96.67 + 0.24
25.58 :: 2.36
Masked diffission
MDM cosine
100.00 +: 0.00
3.67 0.081
0.009
0.001
0.653 1: 0.007
85.96 1: 0.62
69.85 0.41
MDM w/ polynomial
100.00 + 0.00
3.70 0.093
0.010
0.000
0.890 + 0.006
86.57 + 0.55
67.18
0.38
MDM w/ power-law
100.00 - 0.00
3.62 + 0.074
0.007
0.000
0.628 + 0.006
91.30 + 0,54
76.65 +: 0.36
0,0002
0.000
0.947
96.49
ZINC250K
Method
Valid.
FCDJ
NSPDK4
Scaf.1
Uniq.1
Novel.1
Flow bused
MoFlow
63.11 = 5.17
20.93 0.184
0.046
0.002
0.013 + 0.005
0.01
100.00 0.00
GraphAF
68.47 = 0.99
16.02 + 0.451
0.044 0.005
0.067 + 0.016
98.64 = 0.69
99.99 + 0.01
GruphDF
90.61 = 4.30
33.55
0.150
0.177 0.001
0,000 + 0.000
99.63 0.01
100.00 0.00
Continuous diffusion
EDP-GNN
82.97 +
16.74
1,300
0.049
0.006
0.000 + 0.000
99.79 0.08
100.00 0.00
GDSS
97.01 + 0.77
14.66 0.680
0,019
0.001
0.047 + 0.005
99.64 + 0.13
100.00 0.00
GruM
98.65
2.26 0.084
0,0015
0.0003
0.530 + 0.044
99.97 0.03
99.98 0.02
Discrete diffuston
DiGress
94.99 =:
3.48
0.0021 0.0004
0.416 + 0.053
0.01
99.99 0.01
Mad diffisioo
MDM w/ cosine
100.00 : 0.00
25.41 0.023
0.051 +: 0.0003
0.001 + 0.000
99.99 0.02
100.00 = 0.00
MDM w/ polynomial
100.00 - 0.00
26.43 + 0.027
0.053
+ 0.0004
0.001
+ 0.000
99.93 = 0.03
100.00 = 0.00
MDM power-law
100.00
26.09 0.031
0.068
0.001
0.000
100.00 0.00
100.00 0.00
MELD
100.00 = 0.00
1.51 0.047
0.0006 + 0.0001
0.559 + 0.015
100,00 0.01
99.96 + 0.02
Table 12: Property-conditionel generation performance of MELD with standard deviation, averaged over three runs. Note that the standard deviation of the baseline results, which are taken from Liu et al. (2024a), are unavailable.
Quality
Property Aligament
Mecoed
Valid
Covee.T
Divers1
Frxz."
FCD]
Syath.
Og Ponn.]
Ng Pen.
CO2 Pern.
15.05
0.307
0.066
26.45
0.18)
2.176
0.014
1.3083 0.21
1,7735 0.00
0.019
palynomiial
19.11
0.5266033
0.37
0.324
0.01
L6805
1,9446 0.027
0.011
powee-law
17.31
0.401
0.104
0,013
14100
18836
01631 0.141
To empirically verify this, analyze the learned embedding matrix H on benzene ring and compare it to random walk positional embeddings (Dwivedi et al., 2022). In Table 14, we evaluate the average pairwise cosine similarity for (1) node embeddings h (2) edge embeddings h (connected to benzene ring), and (3) all element embeddings (nodes and edges). Our learned embeddings exhibit significantly low pair-wiss wise similarity, suggesting that the learned embedding successfully distinguishes elements even within the symmetric structure.
D.3 ROBUSTNESS ACROSS DIFFERENT DIFFUSION STEPS
We evaluate the performance of MELD on the Polymer dataset under varying diffusion steps, setting the total timestep 7' E {50, 100, 150, 200). Note that during this experiment, we fix the MELD- incorporated MDM to be trained upon a fixed diffusion step of 200, and only vary the number of steps taken during inference. We compare the performance of MELD against that of the strongest baseline, GraphDiT (Liu et al., 2024a), which is originally evaluated at diffusion step size of 500. As depicted in Figure 5, MELD overall exhibits robust performance across range of metrics.
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Table 13: Additional performance comparison MELD with CDGS and JODO on ZINC250K dataset. WD represents Wasserstein distance between the distributions of molecular properties for generated and test molecules.
Method
Total iter.
Diffusion steps
Valid.t
FCDL
NSPDK,
Scaf:1
Uniq:t
Novel.
WD/logPI
WDIQED!
WDISAL
CDGS
2000000
1000
100,00
2.41
0.00077
0.475
99.97
99.99
0.220
0,008
0.292
JODO
1500000
1000
100.00
1.58
0.0004
0.599
100.00
99.94
0.464
0.022
0,380
MELD
263400
500
100,00
1.51
0.0008
0.560
99.99
99.95
0.340
0,004
0.110
100.00
00:00
75.00
60.00
Valle
50.00
0.50
4.00
25.00
0.25
0.00
0.00
0.00
100
Difuuiinn steps
Diffusion steps
DIffusion steps
Diffusion steps
1.00
0.60
9.00
C02 Perm.
0.600
Syitth.
0.40
0.50
0.40
0:20
0.20
0.20
Gruph
Dr
Gr
Griph
0.00
50
100
150
200
50
100
150
200
50
100
150
200
50
300
150
200
Diffusion steps
Diffusion steps
Diffusion steps
Diffusion steps
Figure Performance of MELD under varying diffusion steps. The dotted line indicates the performance of the strongest baseline, GraphDiT, evaluated at fixed diffusion step of 500.
D.4 PER-ELEMENT SCHEDULING OF MELD
In Figure 4, we visualize the variation in per-step learned noise schedules across nodes and edges during the forward diffusion process. Specifically,
Pdge
take 200 samples and plot the vari- ation of the normalized masking probability a, defined as the standard deviation of I-as,
0.35
0.03
We observe consistently higher variance for edge schedules across all timesteps, suggesting that the model prioritizes differ- entiating edges more aggressively than nodes during training. In addition, state-clashing problem inherently intensified in the later steps of the forward process for both nodes and edges, as expected.
0.011
0.2
0.
877
Figure 4: Variation of normalized masking probability a.
D.5 STATE-CLASHING ON SYNTHETIC GRAPH
While the state-clashing can also arise in other discrete graph domains such as ci- tation graphs (Bernecker al., 2024) or social networks (Ji et al., 2024), they typ- ically involve more diverse set of node and edge types, which reduces the like- lihood that distinct graphs collapse into identical intermediate states. However, in synthetic graphs such as SBM (Martinkus al., 2022), which contain only single node type and binary edge types (denoting edge existence), state-clashing is susceptible to occur as depicted in Table 15.
Table 15: Number of unique graph states across varying timesteps in synthetic graph domain (SBM), averaged over 3 seeds.
Method
T-100
T-75
T-50
T-25
T-1
MDM cosine
36.7
11.7
4.3
MDM polynomial
72.0
72.0
58.0
MDM pawer-law
72.0
66.0
MELD
72.0
72.0
72.0
65.7
Using SBM synthetic graph (Martinkus et al., 2022) as representative, we conducted the same quantitative analysis done in Table 6. Due to the high computational cost of applying full graph isomorphism check on the original graphs, we adopted practical approximation: for each of 72 test and validation graphs, randomly sampled 10 nodes and performed the state-clashing analysis.
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T/2 T/4 T/8 T/16 T/32
T/32
Figure 6: More comparisons between element-agnostic power-law scheduling and MELD during reconstruction the ZINC250K dataset. With proposed noise schedule, most reconstruction occurs at earlier timesteps relative to element-agnon -aanonic approach.
This procedure repeated over 3 random seeds ensure consistency. The results show that MELD mostly outperforms standard MDMs in terms of distinguishabilit..
D.6 STATE-CLASHIN ON AARGE-SCALL GRAPH
We further provide state-clashing analy- sis the large-scale Guacamol dataset. Since direct graph isomorphism tests are computationally infeasible quantify the degree of state-clashing, we use approx- imate Engerprint-basee hashing method to count the number of graphs that are provably distinct. Specifically, for each intermediate graph, we compute canon- ical graph fingerprint using a set of graph properties that remain unchanged under node relabeling. The fingerprint is constructed by combining the numbers of nodes and edges, sorted list node degrees, and sorted multisets node and edge types (including masked tokens). These components concatenated into canonical string (e.g., 141c3ld:1, 1.2,2llv:06666.880 identical if their fingerprints match.
Table 16: Number of graphs with distinct fingerprints across varying timesteps in the Guacamol dataset, aged over 3 seeds.
Method
T-100
T-75
T-50
T-25
T-1
MDM w/ cosine
144
144
144
134.3
4.7
MDM w/ polynomial
144
MDM w/ power-law
144
144
144
144
85.7
MELD
144
144
144
144
115.7
1,2) and passed through a SHA-1 hashing. Graphs are counted as
We sample molecules from Guacamol using the maximum graph size in its test set that has sufficient number of samples (> 100), resulting in collecting 144 molecules with 58 nodes each. The results in Table 16 below demonstrates that MELD retains substantially more distinct graph states at the last timestep than baselines, even for larger molecules.
D.7 MORE EXAMPLES OF REVERSE PROCESS
We provide additional visualizations of reverse diffusion trajectories under MELD compared with those from a fixed llwer-law (element-agnostic schedule Figure 6. Consistent with our earlier analysis, MELD achieves faster reconstruction than standard MDMs. For instance, fragments begin
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to unmask as early t = T 1, whereas the element-agnostic schedule only starts recover them at S T/8.
D.8 MOLECULE VISUALIZATION
In this section, we provide 2D visualization of molecules generated by MELD. illustrated, MELD generates chemically realistic molecules for polymers dataset with larger number of atoms (i.e., [V] s 50), verifying its robustness under various graph sizes.
QMB
ZINC250K
Polymer
Figure 7: Visualization molecules generated by MELD.
E DISCUSSION WITH SUBSTITUTION BASED CORRUPTION
In this section, we provide a more detailed discussion comparing our approach with substitution-bast corruption methods. Indeed, substitution based corruption (illustrated Figure 8 (b)) can theoretically produce multimodal target distribution well masking based corruption (depicted in Figure (a)). However, the probability of creating multimodal targets negligible in substitution-bia methods compared to masking- based corruption. Below, we formalize this claim and demonstrate that state-clashing dominant failure mode for MDMs using a toy example.
Formalizing state-clashing with collision probability. Let g denote clean graph and ğ: the noisy graph at time t. Consistent with our manuscript, we denote (e g) as the corruption kernel in the forward diffusion process. We define the state-clashing
the event where distinct clean graphs 91, 92, collapse into the same noisy graph 9i during the forward diffusion process. To quantify this, consider the collision event Ct at time for two distinct, independent graphs gı g2:
Cg 92) (g)
92 gi + g2).
ğI ge(: g))) ge(- g2).
(8)
The severity of the collision is determined by the collision probability Pr(C,(91,92)) Eg 9:(g)
gi (g 92).
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(a) Masking-based noise
(b) Substitution-based noise
(c) Collision probability noise levels
Collzion mrobatiity PTCAAUU 2011
Mekking
Subett di 111--10
(MM-20
Figure 8: (a) Illustration of masking-based noising, and (b) illustration of substitution-based noising. (c) Quantitative comparison of state-clashing probabilities (Pr(C() between masked and substitution- based corruption. The analysis assumes L 23 (average number of nodes in ZINC250K dataset) with k = 2 differing positions. While the collision probability for masking (solid black line) approaches 1 as the noise level + 1, substitution- based methods (dashed lines) exhibit negligible probabilities that decrease further as vocabulary size | increases.
Derivation of collision probability. To compare two corruption approaches, consider simplified scenario where graphs are sequences of nodes of length over a vocabulary V. Each node is specific timestep f. Take two clean sequences Ii and
independently corrupted with probability 92 that differ at exactly k positions and are identical at L - k positions. Note that while graphs are unordered, this clement-wise comparison holds without loss of generality for permutation-cquivarini denoisers.
(1) Case Differing positions (1) + ga(i))
a, otherwise with probability a we replace with single special token [MASK]. A collision occurs only both tokens are masked. Thus, the collision probability Pail is computed as:
Masked corruption: keep the original token with probability -
mask
Pr(ğ (f) (2( g (1) + (2) a
Paiff
Substitution corruption: keep the original token with probability 1 a, otherwise with
probability a we randomly replace it with a token sampled uniformly from V. A collision occurs if one is kept and the other is substituted to that token, (ii) both are substituted to the same token. This leads to the collision probability su as follows:
(i) * gi) = a(1 ax)
a²
a a)
sub Paiff
(t) @2(1)
(10)
M
M
M
(2) Case B: Identical positions (g) (i) g2))))
Masked corruption: A match occurs if both tokens are (i) masked or (ii) kept. Thus, the collision probability massk is computed
Peg
mask
a² + a)-
(1))
Substitution corruption: The corrupted tokens match if (i) both tokens are retained or (ii) both substituted to the same token or (iii) one is kept and the other is substituted to the same token. The collision probability for Pey is:
a(1
(1 -1- a) a)2+ ax a)
sub Pey
(1
a)² +
+
(12)
Vi
D
Assuming independence across positions, the total collision probability is the product over all sites:
(CCC(ggg22))mm. =
(pmnck maskus / (p mask La
Pr(Ce(g1.g2))e = (pat) & suby L-k
(pea
(13)
For positions where two graph elements differ, the ratio of collision probabilities is:
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mask diff
a²
| 2 a
(14)
suh
a(2-a) [V]
This ratio exceeds whenever M++ For a typical vocabulary size in molecular datasets (e.g., [V] 9 for ZINC250K) masked corruption yields a higher collision probability for all a > 0.2. A similar trend applies to identical positions (Pea). Since the term (1 a)² is common to both methods, the inequality is determined by the remaining corruption terms, which follows the same ratio derived above.
Crucially, in the intermediate and later stages of diffusion (where a 1). this gap becomes extreme. As a + 1, the collision probability for masking approaches I, whereas for substitution, approaches (1/DV)) where distinct graphs are highly scattered. That is, lima-+1 Cr (g1.8a)
Pr[Ce(g1.02 (11)).00))mss Jusk 2 |V]
We visualize the gap between Pr|Ce(g1, 92)l musk and (1..((9992))ll in Figure 8 (c). The plot confirms that the gap becomes extreme as a - 1. Specifically, when = 1, the ratio (r[Cr
a 1016. This confirms that while substitution can theoreti- cally produce multimodal targets, the probability of such an event is orders of magnitude lower than masking with fair amount of vocabulary. Thus, state-clashing is much more pertinent to MDMs.
F COMPARISON BETWEEN STATE-CLASHING AND SYMMETRY-BREAKING
Several prior work (Lawrence et al., 2025; Laabid et al., 2025; Wang et al., 2024; Kaba & Ravan- bakhsh, 2022) have introduced the concept of symmetry symmetry-breaking self-symmetric inputs. For instance, Lawrence et al. (2025) and Laabid et al. (2025) identified that equivariant denoiser is unable to yield less symmetric output from highly self-symmetrie noisy input. In this section, we provide two examples where state- clashin and symmetry-breaking formulations clearly differ.
State dashing symmetric Intermediate states
Non-clashing states necessitating symmetry breaking
10
91
Equakorinnt
AAMONE
Figure 9: (a) Illustration of state-clashing between two non-isomers, collapsing into the asymmetric intermediate state. (b) Example case when the equivariant denoiser fails to reconstruct the original molecule from g
(2) although state-clashing does not happen.
We point out that state-clashing can occur even for non-symmetric cases, not only covering self-symmetric inputs. As illus- trated in Figure 9 (a), masking the fluo- rine atom and bond in 2-Fluoroethanol (g) produces the same masked state as masking the chlorine atom and bond in
Table 17: Comparison between MELD and SymPE on ZINC250K dataset.
Method
Valid
FCD.
NSPDK]
Scaf.f
Uniq.1
Novel.
SymPE MELD
100.00
8.24
0.0130
0.261
99.97
100.00
100.00
4.64
0.0044
0.370
99.99
100.00
2-Chlorocthamol ))) The collapsed masked state is not self-symmetric, yet the denoiser still faces multimodal reconstruction target. Our emphasis is mismatch between the joint distribution and the product of marginal distribution for denoising labels, e.g., p(11, x22 * p(æ1) ((2)))(()))))))))))))))))))))))) This can happen even when there exists no symmetry in intermediate states nor the reconstruction targets.
In contrast, symmetry breaking can help even when there exist no state-clashing, which we depict an example in Figure (b). Here, even when state-clashing happens between distinct graphs,
24

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Published as a conference paper at ICLR 2026
the reconstruction will fail for cormmutttioo-qquvvartii neural networks. Thus, state-clashing and symmetry breaking refers to distinct problems arising from lack of expressive power in (a) element- wise independent prediction and (b) permutation equivariant architectures used by the dccoder, respectively. To compare our method with symmetry breaking approaches, we provide a direct empirical comparison against SymPE (Lawrence al., 2025) on the ZINC250K dataset. Following the experimental setup in Lawrence et al. (2025), we adopt the graph transformer denoiser used in DiGress and evaluate both methods under the identical MDM framework. As demonstrated in Table 17, MELD consistently outperforms SymPE across all key metrics. In particular, MELD reduces FCD by nearly 50%.
25


Reviews:
[{"review_id": "F8d63rbq7E", "reviewer": "Reviewer_HAmM", "summary": "This paper proposes a state-clashing problem, meaning that when masked diffusion models are applied to molecular generation, different molecules collapse into the same state, which increases the difficulty of learning. The authors propose an element-wise learnable method that alleviates this issue by learning different corruption rates.", "questions": "1.\tThe authors propose an element-wise learnable method that learns different corruption rates. Is there any suitable case study that has been analyzed to show the relationship between the learned different rates and the various types of elements?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.\tThis paper explores the performance of the currently popular MDM in the field of molecular generation, which is a research topic worth pursuing.\n2.\tThe authors make improvements based on the MDM by introducing element-wise embedding to adapt it to molecular generation tasks.\n3.\tThe authors also validate the effectiveness of the method on large-scale datasets such as Guacamol.", "weaknesses": "1.\tSome overclaims in the paper need clarification, such as the statement that in previous work the transition probabilities between elements in the forward process are all uniformly distributed.\n2.\tWhen proposing the state-clashing problem, the authors lack demonstrations on large-scale datasets. This makes it difficult to convince readers whether such a problem truly exists.\n3.\tThe cases in Figure 2 are not easy to understand and require clearer explanation.\n4.\tSome parts of the method that are based on existing methods could be moved to the appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xLGXjBuEHi", "reviewer": "Reviewer_qGV2", "summary": "This paper investigates the application of Masked Diffusion Models (MDMs) to molecular generation. The authors identify a key limitation in standard MDMs, which they term the \"state-clashing problem,\" where fixed, element-agnostic noise schedules cause distinct molecular graphs to collapse into identical corrupted states during the forward process. To address this, the paper proposes Masked Element-wise Learnable Diffusion (MELD), a framework that learns a flexible, per-element (atom and bond) noise schedule. Through extensive experiments on unconditional and property-conditioned molecular generation tasks, the authors demonstrate that MELD achieves 100% chemical validity on QM9 and ZINC250K and outperforms standard MDMs and other diffusion-based baselines in distributional and property alignment.", "questions": "1. To better situate your work, could you discuss and ideally provide an experimental comparison against more baselines with a fixed schedule?  This is essential to demonstrate the advantages of a learnable schedule over a powerful denoiser with a fixed schedule.\n2. Could you revise Figure 3 for better clarity? The current visualization is difficult to interpret. A clearer format, such as using 2D renderings and highlighting unmasked elements, would more effectively demonstrate the claimed faster recovery of your method.\n3. Could you add a brief discussion of the training overhead (e.g., increased training time) introduced by the learnable scheduling network to the main paper? This would help readers better understand the practical trade-offs of your approach.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Originality and Significance. The paper makes a significant and original contribution by identifying the \"state-clashing problem\" as an obstacle to applying standard MDMs to structured data like molecular graphs. The core idea of learning an element-wise forward process to orchestrate distinct corruption trajectories is an elegant and insightful solution. \n2. Quality. The technical quality of the work is high. The hypothesis about state-clashing is well-motivated and convincingly demonstrated through both theoretical formalization and empirical analysis. The proposed MELD framework is a technically sound and well-designed solution. The experimental evaluation is comprehensive, covering multiple datasets (QM9, ZINC250K, Polymers, Guacamol) and tasks (unconditional, conditional), and benchmarking against a wide range of strong baselines.\n3. Clarity. The paper is well-written and easy to follow. The state-clashing problem is introduced with intuition and supported by clear illustrations. The experiments are well-structured, and the results are clearly communicated through tables and figures.", "weaknesses": "1. The paper's central claim of superiority is undermined by an incomplete set of baseline comparisons. While MELD is shown to be effective against standard MDMs and some diffusion models, it omits a direct comparison to some relevant works. Methods presented in \"Conditional Diffusion Based on Discrete Graph Structures for Molecular Graph Generation\" and \"Learning Joint 2-D and 3-D Graph Diffusion Models for Complete Molecule Generation\" have demonstrated exceptional performance on ZINC250K benchmarks. These models achieve their results using strong denoising architectures but with simple, fixed noise schedules. \nThis raises a critical question: Is the added complexity of learning the forward process truly necessary if a stronger denoising architecture with a fixed schedule can achieve similar results? The paper needs to more explicitly articulate the unique advantages of its approach beyond incremental performance gains. For instance, does MELD offer better parameter efficiency or faster training? Without a clear, compelling advantage, the rationale for introducing a learnable noise schedule, which slows down training, is weakened.\n\n2. Ineffective Visualization in Figure 3. The visualization in Figure 3, which aims to demonstrate MELD's faster recovery during the reverse process, is not very effective. The overlaid text (\"[MASK]\") and molecular structures are small, cluttered, and difficult to parse. This makes it challenging to visually verify the claim that MELD reconstructs meaningful fragments earlier than the element-agnostic schedule. A clearer visualization would be much more impactful.\n\n3. Missing Discussion of Training Overhead in Main Text. The paper introduces a learnable noise scheduling network and a joint optimization procedure, which inherently adds computational cost and complexity during training compared to models with a fixed forward process. However, the main text lacks any discussion of this overhead.  A summary of these findings should be included in the main paper to provide readers with a complete picture.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LGlhwSR1hy", "reviewer": "Reviewer_syMP", "summary": "This paper focuses on the adaptation problem of Masked Diffusion Models (MDMs) for discrete data in molecular graph generation. It points out that \"fixed, element-independent\" forward masking scheduling leads to different molecules collapsing to the same intermediate state in the forward trajectory, making reverse denoising, typically unimodal and predicting independently by node or edge, difficult to learn the correct reconstruction target. To address this, the paper proposes MELD: which learns the forward masking rate at the element level (node/edge) and assigns an independent erosion trajectory to each graph element through a parameterized noise scheduling network; it is jointly optimized with the reverse denoising network during training. The authors claim that MELD achieves high efficiency in unconditional generation of QM9 and ZINC250K graphs and outperforms standard MDM and several diffusion baselines in distribution alignment and property alignment.", "questions": "1.The statement \"first 100% validity\" conflicts with Table 1. It is recommended to change it to \"significantly reduced FCD/NSPDK while maintaining 100% validity.\" Could you please report the confidence intervals for inefficiency (multiple sampling)?\n\n2.Can a more systematic comparison be made between the key differences and complexity of existing \"adaptive/category-level\" scheduling (such as DiffusionBERT, GenMD4, TabDiff) and the \"element-level\" scheduling in this paper? Currently, only a rough comparison is made in Table 3, lacking a theoretical analysis of the differences in expressive power.\n\n3.Please list the number of training epochs, learning rate, scheduler, backbone, number of GPUs, and training time for all baselines; and specify which baselines were retrained by the authors and which were reproduced from the original paper.\n\n4.Please add \"Node relabeling robustness test\" (variance of distribution index/property MAE under multiple labels of the same molecule).\n\n5.Table 5 only performs isomorphism counting on 12 nodes/131 samples, which is costly but has a small sample size; it is recommended to provide estimation methods for larger scales (such as approximate GI or fingerprint hash upper/lower bounds) and statistical confidence intervals.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.An intuitive explanation and formal analysis of the \"state-clashing\" phenomenon are given, pointing out that fixed, element-independent forward occlusion makes it easy for different graphs to fall into intermediate states with poor distinguishability, resulting in a highly multimodal posterior and a model approximating a \"unimodal, decompositional\" distribution, which in turn produces solutions with high entropy and distribution shift. Formulas (3) and (4) are relatively clear with textual explanations.\n\n2.Both unconditional (QM9, ZINC250K, Guacamol) and conditional generation are evaluated; it also includes ablation (fixed vs. learned scheduling, node/edge/node+edge) and the \"number of intermediate states\" metric to characterize state-clashing.\n\n3.The training objective employs CE weighting by node and edge, and the gradient of the discrete sampling is discussed in the pass-through estimation; these are consistent with recent MDM literature.\n\n4.The occlusion rate is learned at the element level to avoid large-scale collisions in the middle time step, and a differentiable ST-Gumbel training path is given.", "weaknesses": "1.The element-level kernel renders the forward process non-equivariant, meaning the intermediate state distribution is affected by vertex permutations. For molecular graphs, this contradicts the fundamental principle that isomorphism should not alter the generative distribution. Current methods merely introduce a learnable embedding H for each graph element and \"randomly permutate columns\" to \"distinguish graph states with the same number of nodes/edges,\" but this does not restore the guarantee of permutation equivariance. It needs to be proven that this forward process, which breaks equivariance, does not induce dependencies on node labels and generalization issues, especially whether relabeling input nodes during testing maintains a consistent sampling distribution.\n\n2.The abstract and main text claim that MELD is \"the first diffusion model to achieve 100% chemigenicity in unconditional generation on QM9 and ZINC250K,\" but several MDM baselines in Table 1 also show 100%. The wording needs to be corrected.\n\n3.The paper does not provide an explicit collision risk function or upper and lower bound analysis; the loss in Equation (3) does not directly minimize the \"collision probability\". It is suggested to provide a computable proxy metric and its relationship with the gradient direction, or to supplement the appendix with a simplified derivation of the \"collision probability as a function of {𝑤_{𝑖}}\".\n\n4.The manuscript states that \"unless otherwise specified, standard MDM and MELD use the same DiT backbone,\" but were the other discrete/continuous diffusion baselines in Table 1 also retrained and had their backbones and training budgets aligned? If comparisons are only made within the MDM family without aligning the backbones/hyperparameters of external distributed models, the conclusions may overestimate the advantages of MELD. Please provide the number of training epochs, GPU configuration, total duration, and FLOPs in the appendix, as well as the retraining/reproduction practices for each baseline.\n\n5.The use of V.U.N.↑ in Tables 3 and 6 lacks a clear explanation of its meaning and calculation in the text (it seems to be a composite score for Validity/Uniqueness/Novelty?). Please define it explicitly at its first appearance in the text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EQQLLiCt5i", "reviewer": "Reviewer_88F6", "summary": "The authors identify state-calshing as a key limitation of Masked Diffusion Models (MDMs), which refers to the way graph components with different labels end up in the same intermediate states when using a unified forward process. This issue\nmakes denoising difficult, especially in the highly symmetrical space of molecular structures. To mitigate this issue,\nthe authors propose state-dependent learned forward processes to collect structural information about the molecule and maitain the partial identity of graph elements during denoising. The paper compares to relevant baselines in diffusion-based molecular generation and shows empirical improvements in some metrics.", "questions": "- How does MELD perform compared to equivariant models using symmetry breaking?\n- Why do you think other corruption methods maintain partial identity through the forward process?\n\n**Typos & nitpicks**\n- Line 39: scalability\n- Line 90: scalability and generalizability\n- Line 255: '... as less probability of state-clashing,...' => incomplete phrase\n- Line 457: '... a higher count of represents' => missing word\n\n## References \n[1] \"Improved Equivariant Networks with Probabilistic Symmetry Breaking\", Lawrence et al., 2025.\n\n[2] \"Equivariant Denoisers Cannot Copy Graphs: Aligns Your Graph Diffusion Models\", Laabid et al., 2025.\n\n[3] \"Discovering symmetry breaking in physical systems with relaxed group convolution\", Wang et al., 2024.\n\n[4] \"Equivariant networks for crystal structures, Kaba et al., 2022.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper identifies a specific limitation in MDMs and addresses it appropriately.\n- The algorithm is clearly described. \n- The main baselines in molecular generation are included in the empirical analysis.", "weaknesses": "**TL;DR** the weakest points for me are the positioning w.r.t the symmetry breaking literature, and the comparison to other noise methods used in diffusion (e.g. uniform, marginal, etc). I am willing to change my score if the author clarify these points and address some presentation issues. \n\n**Connection to the symmetry breaking literature**: I understand that MELD seeks to prevent the state-clashing problem from happening in the first place, but how does it compare to methods using an equivariant forward process and denoiser, and breaking the symmetry of the 'highly similar' intermediate states during denoising? In other words, what is the connection between your work and the broader topic of symmetry breaking as tackled by [1,2,3,4]? I am asking in particular because MELD seems to sacrifice equivariance entirely, I am curious to see how it compares to models using the inductive bias of equivariance (e.g. to model the symmetries inherent to molecules and other graphs) while breaking the symmetry introduced by the forward process (highly similar masked intermediates).\n\n**Claims regarding 'substitution-based corruption methods'**: The authors explain that other noising approaches (I am assuing methods like uniform or marginal noise) suffer from state clashing less than MDMs. While noising with random labels *looks* less similar than noising with the same mask token, uniformally generated intermediates should have the same amount of information as masked intermediates (i.e. in the o-Phenylenediamine and m-Phenylenediamine molecules, a uniform noise schedule should struggle as much as MDMs with recovering the right isomer). I find the claim that such methods 'preserve structural similarity and retain partial identity' unsubstantiated. Can you elaborate more on this? \n\n**Presentation**: \n- Since the state clashing problem is the main motivation of the paper, it would be more natural to present it first in the methods section and to emphasize it more in earlier sections (introductions, related work).\n- Some claims in the paper are exagerated. For example, saying that applying diffusion to molecules is \"underexplored\" (abstract and introduction line 51), or that MELD achieves notable empirical improvements over the baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)