GAGA: Gaussianity-Aware Gaussian Approximation for Efficient 3D Molecular Generation
Authors: Jingxiang Qu
Wenhan Gao
Ruichen Xu
Yi Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Gaussian approximation
trajectory truncation
efficient generation
3D molecular generation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Gaussian Probability Path based Generative Models (GPPGMs) generate data by reversing a stochastic process that progressively corrupts samples with Gaussian noise. Despite state-of-the-art results in 3D molecular generation, their deployment is hindered by the high cost of long generative trajectories, often requiring hundreds to thousands of steps during training and sampling. In this work, we propose a principled method, named GAGA, to improve generation efficiency without sacrificing training granularity or inference fidelity of GPPGMs. Our key insight is that different data modalities obtain sufficient Gaussianity at markedly different steps during the forward process. Based on this observation, we analytically identify a characteristic step at which molecular data attains sufficient Gaussianity, after which the trajectory can be replaced by a closed-form Gaussian approximation. Unlike existing accelerators that coarsen or reformulate trajectories, our approach preserves full-resolution learning dynamics while avoiding redundant transport through truncated distributional states. Experiments on 3D molecular generation benchmarks demonstrate that our GAGA achieves substantial improvement on both generation quality and computational efficiency.

Published as a conference paper at ICLR 2026
GAGA: GAUSSIANITY-AWAE GAUSSIAN APPROXI- MATION FOR EFFICIENT 3D MOLECULAR GENERA-
TION
Jingxiang Qu', Wenhan Gaoi, Ruichen Xu², Liu
Department of Computer Science, Stony Brook University
2 Department of Applied Mathematics and Statistics, Stony Brook University
ABSTRACT
Gaussian Probability Path based Generative Models (GPPGMs) generate data by reversing stochastic process that progressively corrupts samples with Gaussian noise. Despite state-of-the-art results in 3D molecular generation, their deploy- ment is hindered by the high cost of long generative trajectories, often requir- ing hundreds thousands of steps during training and sampling. In this work, we propose principled method, named GAGA, to improve generation efficiency without sacrificing training granularity or inference fidelity of GPPGMs. Our key insight is that different data modalities obtain sufficient Gaussianity markedly different steps during the forward process. Based on this observation, we ana- lytically identify characteristic step which molecular data attains sufficient Gaussianity, after which the trajectory can be replaced by a closed-form Gaussian approximation. Unlike existing accelerators that coarsen reformulate trajec- tories, our approach preserves full-resolution learning dynamics while avoiding redundant transport through truncated distributional states. Experiments on 3D molecular generation benchmarks demonstrate that our GAGA achieves substan- tial improvement on both generation quality and computational efficiency. The code is available at https: : //tithuu..././////// com/ Http///uXX/GGAAA
INTRODUCTION
Gaussian Probability Path based Generative Models (GPPGMs) refer to a family of generative mod- els which define a probability path that smoothly transforms simple Gaussian distribution into the target data distribution (Lipman et al.) (022). Recently, they have demonstrated impressive per- formance across diverse domains such as images (Li al., 2019), text (Austin et al., [02T), and molecules (Zhang et al.] (2023). However, the generative trajectories are typically modeled as so- lutions a stochastic differential equation (SDE) or ordinary differential equation (ODE); such solutions are often discretized by hundreds to thousands of steps for better learning granularity. The heavy computational demand thus becomes one of their key limitations, especially for 3D molecu- lar generation. To improve the efficiency, prior work has largely focused on sampling acceleration, for example, coarsening trajectories with reduced-step solvers (Song et al.] 2020 Lu et al. 2022] Karras et al. (022) and retrieval-based methods (Zhang al. (025). While effective for inference, these approaches either compromise trajectory granularity or leave training costs unaffected. Efforts closer to training, such as adaptive priors (Lee et al. 2021 Vignac et al. (022) and leapfrog initial- izers for trajectory prediction (Mao et al.) 2023). still depend on modifications of the noising process or specialized architectures, rendering them domain-specing and difficult to apply 3D molecular generation.
In this work, we propose novel method that improves both training and sampling efficiency of GPPGMs via Gaussian Approximation (GA). Motivated by the translational invariance of molecu- lar data, we leverage this unique property to design GAGA, which operates directly on zero-mean invariant data without any loss of structural information, as analyzed in 2.2 Rather than coars- ening the generative trajectories or modifying the predefined noise schedule, our method identifies
"oorresonndence yi.liu.4 stonybrook.edu

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Published as a conference paper at ICLR 2026
characteristic time step T which the input data distribution has effectively obtained sufficient Gaussianity, Based on this point, the generative trajectory can be truncated, and the final distri- bution can be approximated by tractable Gaussian reference distribution with analytically de- rived mean and variance, as shown in Fig. 7 This design yields two key merits absent in existing GPPGMs acceleration methods: (1) the ability for training acceleration via eliminating ineffec- tive optimization on over-noised inputs, and (2) sampling fidelity improvement by maintaining the granularity of the original generative trajectories and focusing more on steps with ad- equate data information guidance during learning. We empirically validate our method across different 3D molecular datasets, demonstrating significant improvements in both sampling and train- ing efficiency with high-quality molecular generation.
Non Zero-mean
Backward
TT
Data Distribution
0
Lt
Forward
(a, kt)
Gaussian
Zero-mean
Invariant
Gaussian Approximation
Backward
The flowehart of the GAGA, where the forward process is discretized to T steps. In such a case, when the noised data distribution æ; has attained sufficient Gaussianity timestep t, we approximate it with reference Gaussian (ir, ūz). Therefore, the length of the generative trajectory can be reduced from T steps steps.
Figure 1:
PRELIMINARIES
2.1 GAUSSIAN PROBABILITY PATI BASED GENERATIVE MODELS
GPPGMs define a probability path g(x:). 1 E [o, 10 where the trajectory smoothly transforms a simple Gaussian prior into the target data distribution. At t O, ((xo) is the data distribution; at = 1, ((x1) is standard Gaussian. This probability path perspective gives a continuous in- terpolation between noise and data. Specifically, these models include the diffusion models (Ho et al. 2020 [Song et al. 2021 2020) and the Gaussian flow matching models (Lipman et al.] 2022 Gao al., 2025
While the probability path captures the evolution of the marginal dis- iribttinn q(x,), E [0, I,, more concrete characterization arises from the conditional probability
path q(a;, mo), E (0, I]. which traces the transformation individual data samples. This path defined as
q( 2o) = (xx; Vārzo, T.I).
where a, E [0, with ag = controls the decay of the signal power over time and o. v1 the level of injected noise. Typically, ā; monotonically decreasing so that āj R 0, and xı is very close to the pure Gaussian distribution. This conditional Gaussian probability path is used in denoising diffusion models, and it can be shown that the flow trajectories Gaussian flow matching models also follow the conditional Gaussian probability path (Gao al. 2025 Ma et al. 2024).
Without loss of generality, throughout this paper we assume that the forward process of GPPGMs is pariance-preserving (VP). Similar conclusions can be readily extended other forward settings.
Learning the Reverse Process While the forward conditional path in equation describes the forward noising process from data to Gaussian noise, generative modeling requires the reverse path,
'We follow the convention in prior works, which simplify qe (æx) g(x:). In practice, the continuous time interval is discretized into 7 steps; we use E (0, 1,..7} when referring to discretized implementation. In this work, primarily focus denoising diffusion models and Gaussian flow matching models, which represent the mainstream frameworks within GPPGMs.
2

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which maps Gaussian samples back the data distribution. Given equation closed-form expression for the reverse process:
derive
(at-Ar Ae (2, = - (ai- (T),
(2)
where
He (e) V0t- At ats)
v Val -a--
20 +
24
1-
0. (1 (1)) (1 Cr a1-Ar)
where the ar is the discretization step-wise coefficient, defined by 0z ăe-ar" Consequently, the 2o is typically unavailable in the reverse process. Therefore, the objective of GPPGMs to approximate the reverse process with a parameterized model @ that only conditions on the observable noisy sample: ps (#r- (re) = p (Zr-At zo). To accomplish this, for diffusion models, we minimize the KL divergence between them: KL(p (I:- | It. 2o) || (Iz- Ai Iz)); since both are assumed to be Gaussian, the KL divergence can be simplified to MSE between the means:
ce-ar. ar II. as-
1 (((e)) - pe( 20) 11 (()) - Eyg [ 20?
For Gaussian flow matching models, the training objective is derived from ODE perspective; nevertheless, it can be proven that there is an equivalence for both the training objectives and learned models between diffusion and flow models (Gao et al. 2025). assuming suitable noise schedule. As result, the GA techniques and mathematical insights provided in this work can be effectively applied to both diffusion models and Gaussian flow matching models, demonstrated by the experimental results in Sec. 4.2
2.2 ZERO-MEAL INVARIANCE
A data modality is zero-mean invariant if centering each sample by subtracting its empirical mean preserves all the information necessary for downstream modeling. Formally, let x E RR denote data sample, and define its centered version as:
1ds
(4)
where 1a E Rd is the vector of all 1-s. A data modality is said satisfy zero-mean invariance if, for all æ in the support of the data distribution p(x), the transformation æ + ø retains the semantic structural information of the original input.
This property common in domains where only internal relationships among dimensions carry information, while global offsets are irrelevant redundant. Typical examples include any repre- sentations defined up to an affine baseline or possessing a shift-symmetric structure, such as con- figurations invariant global alignment, label encodings invariant additive bias, or features em- bedded in contrastive spaces. We provide some detailed examples and the corresponding analysis in Appendix Zero-mean invariance permits generative models to operate in a reduced subspace orthogonal to the mean direction, eliminating redundant degrees of freedom. In 3D molecular data, zero-mean invariance is widely employed due to its translational invariance (Hoogeboom et al.,
2022
Hong et al. 2025 Xu et al. 2023).
3 GAUSSIANITY-AWARE GAUSSIAN APPROXIMATION
Building on the preliminaries, we now introduce our framework for shortening the generative trajec- tory in GPPGMs via truncation. Rather than executing the full generative trajectories, we identify a characteristic timestep 7* at which the data effectively exhibits sufficient Gaussianity") This enables an analytic truncation, whereby the redundant trajectory is replaced with a direct Gaussian approx- imation (GA). It significantly improves both computational efficiency and generative fidelity by
Following (Berg et al. 2012). in work, Gaussianity refers to dataset random variable distributed according Gaussian distribution.

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Published as a conference paper at ICLR 2026
enabling the GPPGM skip the optimization over-noised generation trajectory segments while focusing more on the segments with adequate data information guidance the learning process. is noted that since both diffusion and flow- Now-matching models share the same Gaussian probability path, as introduced in Sec. 2.1 GA is equivalently applicable to both for trajectory truncation. For simplicity, follow the setting of diffusion model (Hoogehoom et al. 2022) on the forward process and present our method in this context.
3.1 GAUSSIAN APPROXIMATION
GA is commonly employed in statistics represent intractable conditional or marginal distribu- tions (Berry [941; Deng & Zhang 2020 Chernozhukov et al. 2013). This modeling choice facili- tates closed-form expressions for critical quantities, including transition densities, posterior distribu- tions, and variational bounds, which are essential for both optimization and sampling procedures. In GPPGMs, the forward process can be interpreted as progressively pushing the data toward a Gaus- sian distribution. As illustrated in equation the marginal and transition densities of the trajectories at any finite time index remain Gaussian. However, the intractability of data distribution prevents us from directly calculating the mean and variance of the Gaussian at intermediate timesteps.
Meanwhile, for data modalities that are zero-mean invariant, such as molecular coordinates, point clouds, categorical embeddings, the difficulty of estimating mean can be avoided by enforcing zero-centering as a preprocessing step. Such centering preserves structural information and symme- tries (e.g., translational invariance) (Hoogeboom et al. 2022), while consistently ensuring the mean equals to O, as analyzed in Appendix
In addition, the variance remains intractable obtain exactly. In this paper, estimate it through the per-sample statistics. Given dataset D (x(0) Jiz with (+)) R2, we compute
(
where
(5)
d
j=1
and aggregate across the dataset to obtain the average per-sample variance as é = ΣE (0) This estimator unbiased under mild moment conditions (Vershynin) (2012), and we empirically verify that it's 's an effective estimator for GA in GPPGMs. Therefore, under the VP forward process on zero- zero-meaned data, we plug in i into Equation | as the variance of data samples, then the mean üu and variance é of noised data at intermediate timesteps can have the following analytic form:
ju 0,
1 ā.(1-
(6)
Consequently, for zero-meaned data, once sufficient noise has been injected at timestep T", the marginal distribution æT. can be approximated by (0 Ür. 1). This serves as the foundation for our strategy of trajectory truncation.
3.2 GAUSSIAN APPROXIMATION AND INITIAL DATA DISTRIBUTION
The analysis above shows that, once sufficient noise is injected, the forward process admits a tractable GA. Nevertheless, the following question arises:
(Q) How do determine T* at which the injected noise becomes sufficient for this approxi- mation? Is it the same across different tasks?
To answer this question, we first present Proposition 3.1 to show that T is related to properties of the initial data distribution.
Proposition 3.1. Given L E [0, T) and K 2 3, and the Gaussianity evaluation functional
H(K)
(Cov(E)) + Σ wk ((()
where 14, is the indicator function, 3 > and uk > 0 (k 2 3). D = (Diag(v) E Rd) is the diagonal subspace and
(DD) Diag(diag E);
mp (E) Σ (T))
(8)
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the orthogonal projections. Cou(.) and (X) are the covariance calculator and the k-th cumulant tensor, respectively. Let A, be two initial data distribution, where
H(m) (x & ( for all m ........ K
(9)
holds with at least strict inequality. Then for every
(xx) < (xx))
Consequently, for every e O,
TA = in > t: H (xx)) { < inf(s > H(() { TB
(10)
A formal proof is provided in Appendix This proposition establishes that if the initial data distribution is inherently closer to Gaussian, then the corrupted samples achieve sufficient Gaussianity earlier, and the corresponding GA timestep T be smaller, shown in Fig. 2] In particular, sparse molecular coordinates around equilibrium are closer to Gaus- empirically quantified in Appendix [ In such case, the GA of molecular data can start at a smaller T Since different initial data distribution induces different GA timesteps, we need principled way identify the precise 7 Therefore, in Sec. 3.3 de- velop statistical Gaussianity evaluator that serves as an operational test, combining dependency measurement and distributional similarity criteria to precisely identify the GA timestep T"
qualitatively
sian (Frenkel & Smit 2023),
Raw data
T= T50
T 450
T
T
(a)
Data Distribution
Gaussianity of Image Data in Forward Process
Gaussian
(b)
(c)
Data Distribution
Gaussianity of Molecular Data in Forward Process
Gaussian
Figure 2: Comparisons of the forward noising process across different data modalities. (a) shows a continuous-valus image matrix, while (b) and (c) illustrate the distribution of molecular data consisting of one-hot vectors for atom types and 3D Euclidean coordinates for atom positions, re- spectively. The same schedule for gaussian probability path is applied across all modalities, with the number of forward timesteps T up to 1000. Despite identical signal-to-noise ratios, molecular data obtains sufficient Gaussianity for significantly fewer steps compared to image data. This comes from the different Gaussianity across initial data distributions, empirically quantified in Appendix
3.3 EVALUATING GAUSSIANITY: DATA DEPENDENCY DISTRIBUTIONAL SIMILARITY
While the preceding analysis suggests that 2; may be approximated by a Gaussian, the validity of this approximation fundamentally depends on whether the æT- has gained sufficient Gaussianity for GA. In this section, we present the Gaussianity evaluation method from the perspectives of data dependency and distributional similarity.
Data Dependency Decay. The timestep at which data loses its structural information under pro- gressive noise perturbation is critical for establishing the valid Gaussian approximation. Since the Gaussian distribution in GA is independent, the disappearance of data dependency in 2, is suf- ficient condition to the independent Gaussian approximation. As shown in Fig. 2 under the same

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noise schedule, the timestep at which the data has obtained sufficient Gaussianity strongly depends on the underlying data modality. 3D molecular data combines one-hot atom types and 3D Euclidean
coordinates (Liu et al. 2022; Wang et al. 2022; Yan al. 2022 Wang al. 2023 Lin et al. 2023 Subedi et al. 2024 Liu et al. [025] Qu et al. 2025). Due to sparsity and low dimensionality. the initial molecular data distribution preserves less dependency. In contrast, natural images
high-dimensional and spatially correlated, with smooth local structures that preserve structural de- pendeney over many more noising steps. Monitoring the decay of data dependency thus provides principled criterion for determining the characteristic timestep T at which Gaussian approximation becomes valid.
Consequently, to quantify the Gaussianity from the perspective of data dependency, we adopt the mutual information (MI) test (Kraskov et al.) 2004) as our evaluation functional. Because exact independence occurs only at the terminal prior æt ~ (O,,,), I), we adopt tolerance Edep >> 0 and define the dependency- timestep as
TDD min
{ Dep(æ,) S Edep J.
(11)
where Dep(-) denotes the MI-based dependency evaluator. Specifically, Dep(#;) is computed as the averaged mutual information across both feature-wiss and component-wiss slices of xt, small value of Dep( indicates that the is statistically independent. It thus provides a concrete condi- tion under which GA becomes valid from the perspective of data dependency. The implementation details of Dep(x..) are provided in Appendix F.1
Distributional Similarity. While data dependency decay captures the disappearance of depen- dence, GA also requires that the marginals of align with those of a Gaussian distribution. To assess this, we measure the distributional similarity between 3; and a reference Gaussian with matching variance using the Kolmogorov-Smimoe (KS) distance (Massey Jr, 1951). Concretely, for each dimension ot we compare its empirical cumulative distribution function (CDF) (r.( with the Gaussian CDF &5, (Ir), and average across all dimensions:
D. = E.EDS Dij. where Di, = sup (z)(()) SupFFF))) j=1
Dz. (x)
(12)
x
A smaller D, indicates closer aligmment with Gaussian marginals and therefore stronger justification for approximation by (ūr, i,f). Since exact convergence only holds at the terminal prior æ₁ NO, I), we adopt tolerance eDS > 0 and define the distributional-simina timestep as
Tos == min {
D. S EDS
(13)
Estimation of As established in Proposition 3.1 molecular data inherently distributes closer to Gaussian. From the joint perspectives of dependency decay and distributional similarity, we provide concrete and quantitative characterizations of the Gaussianity of æ,, ensuring that the approximated 2y- is sufficiently independent and marginally Gaussian. Accordingly, we define the operational GA timestep as
T = max( TDS),
(14)
which guarantees that œ-- satisfies both independence and marginal Gaussianity. The pseudo code of estimation algorithm is shown in Appendix D
Merits of GAGA. Overall, GAGA provides three notable merits: (1) It improves computational efficiency of GPPGMs in both training and sampling by truncating the trajectory after the GA timestep T thereby avoiding ineffective optimization and redundant inference on over-noised data. (2) It enhances generation quality owing to the removal of redundant trajectory segments without meaningful data information guidance, i.c., they have obtained sufficient Gaussianity. This strategy allows the model to concentrate reconstructing the probability path in regions where structural information is still preserved.
(3) GAGA is compatible with existing SDE/ODE solver-based ac- celerators such as DDIM (Song et al. 2020), and also improves generation quality when combined with them, as empirically validated in Sec. 4.4

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4 EXPERIMENTS
In this section, empirically evaluate the proposed method standard molecular generation benchmarks. We present the experimental setup, define the evaluation metrics, and report quan- titative results on both generation quality and efficiency. Additional details on the Gaussianity tests and experimental configurations are provided in Appendix | and Appendix G respectively.
4.1 EXPERIMENTAL SETUP
Datasets. We conduct experiments on widely-used molecular datasets, QM9 (Ramakrishhan et al., 2014) and GEOM-Drugs (Axelrod & Gomez-Bombarellij (002). QM9 contains 130K small molecules with up to 29 atoms, while GEOM-Drigs comprises 450K drug-like molecules with an average 44 and up to 181 atoms. The configuration of datasets follows Hoogeboom et al. (2022) for regular generation and Xu et al. (023) for latent-space generation, respectively.
Baselines. We conduct extensive experiments to compare with several state-of-the-arr baselines. For fair comparison, we restrict our baselines to those that do not explicitly incorporate bond in- formation, either in data modeling (e.g., Irwin et al. (2025)) in post-hoc processing (e.g.. (eBylee cct al. (2011). Specifically, G-SchNet (Gebauer et al. 2019) and Equivariant Normalizing Flows (ENF) (Garcia Satorras al. 202T) employ autoregressive models for molecule generation. Equiv- ariant Diffusion Model (EDM) (Hoogeboom et al. 2022). Geometric Latent Diffusion Model (Ge-
oLDM) (Xu et al. 2023), and Equivariant Flow Matching model (EquiFM) (Song et al., (023) are three representative GPPGMs from different perspectives for molecular generation, including reg- ular diffusion, latent diffusion, and flow-matching. Moreover, the variants of EDM (GDM) and GeoLDM (GraphLDM) are also employed for comparison.
Metrics. We evaluate our method on standard molecular generation benchmarks using two broad classes of metries: generation quality and efficiency. For generation quality, we report validity (the proportion of chemically valid molecules according to standard valency checks), uniqueness (the proportion distinct molecules among generated samples), molecular stability (the fraction of generated molecules satisfying correct valency constraints), and atom stability (the fraction of generated atoms satisfying correct valency constraints). Following prior works (Xu al., 2023 Hong et al. 2025), these metrics are computed via the cheminformatic tool RDKit (Landrum et al. 2016) over 10,000 generated samples. For efficiency, we record the average sampling time (S- Time) in GPU seconds per sample and total training time (T-Time) in GPU days. Both of them are measured on identical hardware across different baselines. Moreover, the trajectory length (Steps) T is also shown in the results. These metrics collectively quantify the fidelity, diversity, and practical computational benefits of our method.
4.2 QUANTITATIVE PERFORMANCE
Table 1: Quantitative results on the QM9 dataset. The results of all baselines are directly obtained from their original works. The results of backbones applied with GAGA are highlighted with grey background. We run the evaluation 3 times and report the mean and standard deviation (std). Com- pared with previous methods, GA benefits all methods, achieving up to a 3.6% improvement in the molecule stability metric, and significantly reducing the generative trajectory length by 40%.
Generation
Performance
Effiiency T-Time (GPU day)
Atom Sta (%)
Mol Sta
Valid
Valid Uniq
S-Time (GPU sec.)
Traj. Len. (Steps)
Model
(%)
Data
99.0
95.2
97.7
97.7
ENF
85.0
4.9
40.2
39.4
G-SchNet
95.7
68.1
85.5
80.3
GDM-AUG
97.6
71.6
90.4
89.5
0.52
2.9
1000
GM
97.2
70.5
83,6
82.7
0.36
5.7
1000
EDM
98.7
82.0
91.9
90.7
0.65
5.6
1000
EDM GAGA
98.9 1 0.03
85.6 + 0.22
94.7 = 0.04
92.0 = 0.12
0.36
3.1
550
Gc
98.9
89.4
93.8
92.7
0.64
11.7
1000
GeoLDM GAGA
99.2 0.01
92.3
0.06
96.7 0.14
94.4 0.21
0.42
7.2
EquiFM
98.9
88.3
94.7
93.5
0.17
6.2
200
EquiFM GAGA
99.0 + 0.02
91.2 = 0.11
96.2 = 0.09
93.7 = 0.18
0.15
4.9
160
denotes invalid or not recorded setting in the original publication.
We evaluate the effectiveness of GAGA across multiple molecular generative baselines on both the
QM9 and GEOM-Drugs datasets. The results on QM9 reported in Table
We observe from

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the table that GAGA reduces both training and sampling costs by shortening the diffusion trajectory by up to 40%, and consistently improves generation quality. GAGA does not alter the granularity of the original generative path; rather, it leverages the early timestep of molecular data for obtaining sufficient Gaussianity, which enables the safe trajectory truncation. Therefore, the GPPGM could begin training and sampling from earlier noise step without violating the probability path of the preserved segments. It also enables the model to focus on learning reconstructing trajectories where data information still preserved, thereby improving the final generation quality.
Table 2: Quantitative results on GEOM dataset. We run the evaluation 3 times and report the mean and std. In general. GA improves generation performance and provides better efficiency across models. The results of backbones applied with GAGA are highlighted with grey background.
Generation Performance
Efficiency
Model
Atom Sta (%)
Valid (%)
S-Time (GPU sec.)
Traj. Len. (Steps)
86.5
GDM-AUG
77.7
91.8
1000
GraphLDM
76.2
97.2
1000
EDM
81.3
92.6
10.9
1000
EDM GAGA
84.3 + 0.27
93.4 1 0.49
6.4
650
GeoLDM
84.4
99.3
10.2
1000
GcoL.M GAGA
85.9 + 0.08
99.3 + 0.03
7.9
denotes the invalid or not recorded setting in the original publication.
Similar benefits are observed on the more challenging GEOM-Drugs dataset. In this experiment, following prior work (Hoogeboom et al. [022]] Xu et al.) 2023 Lipman et al. 2022). we omit met- rics such as uniqueness, as consistently remains close to 100% across all baselines. Similarly, we exclude molecule stability, which stays near 0% for all baselines. In general, GAGA consistently improves generation quality and reduces per-sample generation time across various baselines, as shown in Table 2 These improvements on both generation quality and efficiency are partic- ularly noteworthy given that GAGA requires no changes to model architecture. Instead, modifies only the generative trajectory by leveraging the early timesteps for obtaining suffi- cient Gaussianity of molecular data.
4.3 ABLATION STUDY
Table presents an ablation study on the choice of GA timestep 7". We evalu- ate models with both shorter and longer truncation points to investigate the impact of different T" We observe that apply- ing GA too early (T* 450 for EDM and T = 550 for GeoLDM) results efficiency gains but sacrifices the gener- ation quality, since premature truncation collapses residual trajectories maintaining chemical plausibility and structural diver- sity, which are still beneficial for recon- struction of probability path. On the other hand, setting T too late (T* 650 for EDM and T 750 for GeoLDM) hurts generation quality while incurring unnec- essary computational cost, since the data has already attained sufficient Gaussianity and additional steps mainly add redundant noise. These results suggest that the optimal T lies at the earliest point where Gaussianity is achieved, and truncation there preserves generation quality while offering the most substantial efficiency improvement.
Table 3: Ablation study on the GA timestep T* We evaluate EDM and GeoLDM models under longer/shorter trajectories trained on QM9 dataset. We run the evaluation 3 times and report the mean value. The results with estimating 7 via our proposed method are shown with grey background.
Valid Uniq
S-Time
Model
(%)
(GPU sec.)
EDM (7" 1000)
90.7
0.65
EDM GAGA (T 450)
91.4
0.32
EDM GAGA (T" 650)
91.6
0.45
EDM GAGA (T" 550)
92.0
0.36
GcoLDM (I" = 1000)
91.9
0.64
GeoLDM + GAGA (T" 550)
93.3
0.36
GeoL.DM + GAGA (7" 750)
93.5
0.49
GeoLDM GAGA (T" 650)
94.4
0.42
4.4 COMPATIBILITY WITH OTHER ACCELERATORS
Most existing acceleration algorithms for GPPGMs improve sampling efficiency by design- ing an efficient SDE/ODE solver applied in the backward process. Notable examples
clude DDIM (Song et al. 2202)), DPM-Solver Series (Lu et al.] 2022] Zheng et al.) 2023 Lu
al. 2025), Gaussian-mixture solvers (Guo et al. 2023), etc. Since DDIM is the pioneering
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and most widely adopted mem- ber of this family, we take as a representative study the com- patibility of GAGA with opti- mized differential equation (DE) solver-based accelerators. Ta- ble 4 reports results for various backbones under DDIM with 2x acceleration and in combi- nation with GAGA. In addition to the generation improvement on backbone models, the exper- imental results also demonstrate that our GAGA is fully compatible with DE solvers. Compared with relying solely DDIM, the joint configuration of DDIM and GAGA yields improved generation performance while inherit- ing efficiency benefits from both. Moreover, thanks to the preserved granularity of the generative trajectory, GAGA reduces the number of timesteps to level comparable with DDIM, yet substan- tially outperforms as the coarsened trajectories induced by DDIM deteriorate generation quality These results confirm that GAGA is orthogonal to other DE solver- accelerators and can be seamlessly combined with them, while further enhancing generation quality. Impor- tantly, GAGA also improves training efficiency, which is generally unattainable for most DE solver-based accelerators.
Table 4: Compatibility study of GAGA with DDIM (2x acceler- ation). We evaluate them 3 times and report the mean value on QM9 dataset. GAGA-applied backbone models are highlighted with grey background.
Backbone
DDIM GAGA
Valid Uniq (%)
Traj. Len. (Steps)
x
90.7
1000
X
92.0
550
EDM
83.7
500
83,9
275
X
91.9
1000
94.4
650
GeoLDM
85.8
500
87.5
325
5 RELATED WORK
Probability Path based Generative Models (PPGMs). PPGMs generate samples over data dis- tributions by learning a transport process that maps simple prior distributions to complex data dis- tributions through sequence of structured transformations, i.e., the probability path. Specifically, diffusion-base bassid generative models simulate this sequential transformation via SDEs, which have emerged powerful paradigm for multi-modal data synthesis (Croitoru al. 2023 Kementzidis et al. 2025 Xu et al.). However, their iterative sampling (often requiring hundreds of steps) poses significant speed bottleneck. variety of techniques aim to accelerate diffusion sampling, such as progressive distillation (Salimans & Ho 2022) and learned noising schedules (Williams et al.) 2024]. Nevertheless, the training process still typically requires hundreds of steps. Beyond diffu- sion models, Flow Matching offers a fresh perspective on acceleration. Flow Matching models train a continuous normalizing flow by regressing an optimal vector field along prescribed paths. Lip- man et al. 2022) showed that using diffusion-style Gaussian paths in flow matching yields more robust training and faster ODE-based sampling. However, the nonlinear and high-curvature nature of learned transport fields makes it challenging accurately approximate such trajectories with few discretization steps during training (Hassan et al. 2024 Bijkelboom et al. (2024).
Gaussian Approximation (GA). GA has long been cornerstone in machine learning theory and practice. The Central Limit Theorem provides a classical justification: aggregates of many random factors tend toward a Gaussian distribution, which often explains why high-dimensional features or latent codes appear approximately normal (Hazra et al. 2021 Düker et al., 2024). Some researchers have explored the potential of the Gaussian approximation
generative modeling. For instance, Wang & Vastola observe that at high noise levels, the learned diffusion score can be well- approximated by a linear Gaussian model. Therefore, they can skip 15-30% of the sampling steps without degrading output fidelity. Such findings reinforce the idea that Gaussian assumptions can serve as an effective proxy for complex distributions in certain regimes, providing practical speedups without significant fidelity loss.
Relation with Prior Works. Improving the quality-efficiency tradeoff in generative molecular modeling has been approached from several directions. One line of work focuses on designing better
'Notably, we found higher-order accelerators, such as Lu et al. 2022 2025), yield much worse performance than DDIM on molecular generation, because their assumption of smooth higher-order continuity makes them push the noised states toward similar trajectories, which significantly reduces molecular uniqueness.

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Published as a conference paper at ICLR 2026
forward process and understanding SNR behavior, including cosine schedules (Nichol & Dhariwal
2021.
courier-space schedule (Falck et al. 2023). ELBO-based augmentation (Kingma & Gao, and the unified perspective under ODE modeling (Gao et al. 2025). Another line proposes
2023
new generative trajectories tailored to molecular geometry, such as equivariant ODE flows (Song
al. 2023), Bayesian flow networks (Song et al. 2024), and straight-line diffusion (Ni et al.. 2025
While these methods modify the forward or backward probability path to accelerate sampling or improve fidelity, our GAGA addresses the same goal from different angle: keep the original schedule and generative dynamics unchanged, and instead identify when the noised distribution has already reached the Gaussian regime so that redundant high-noise segments can be safely truncated. This design makes our method compatible and complementary to existing approaches. For instance, EDM (Hoogeboom et al., 2022) already adopt optimized noise schedule in forward process and EquiFM (Song et al. (003) successfully reduced the trajectory to much fewer steps. However, our GAGA still improves generation quality while further obtaining better efficiency, as empirically
verified in Sec. 4.2
6 CONCLUSION AND FUTURE WORK
In this work, we introduced a principled framework, GAGA, for efficient GPPGMs 3D molecular generation. By leveraging zero-mean preprocessing and empirical variance estimation, we proposed an analytic Gaussian approximation that identifies characteristic time step T" which the noised data obtained sufficient Gaussianity. This approximation enables the truncation of redundant gener- ative trajectories, which are inefficient transport between "Gaussian-like" distributions. As result, our GAGA can improve the efficiency of both sampling and training with consistent improvements in molecular generation quality.
Limitations and Future Works. Despite its empirical success, current GAGA assumes that the data modality is zero-mean invariant. While this assumption holds many geometric and categor- ical domains, it is not valid for modalities like natural images videos, where the absolute mean carries semantic information. Extending our methodology to such domains requires new techniques for determining Gaussianity without relying zero-mean centering. In the future, desire to de- sign unified framework of GAGA, which is capable of accommodating both zero-mean invariant and non-invariant modalities. It would significantly enhance the generalization capability of GAGA in probabilistic generative modeling. In addition, our Gaussianity evaluator may also serve as a tool for reallocating timesteps in quality-focused settings, offering complementary research direction beyond acceleration.
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ETHICS STATEMENT
This work introduces a methodological contribution for improving the efficiency of 3D molecu- lar generation models. All experiments are conducted on standard public benchmarks (QM9 and GEOM-Drugs) without sensitive or personally identifiable data. While molecular generative models have potential applications in drug discovery and material design, such use requires careful domain- specific safety and ethical oversight. Our contribution is limited to algorithmic efficiency and eval- uation on benchmark datasets, and does not involve deployment to real-world systems.
REPRODUCIBILITY
We provide full details of our method, including theoretical derivations, Gaussianity evaluation pro- cedures, and implementation settings, in the main text and appendices. All experiments are con- ducted on publicly available datasets (QM9 and GFOM-Drugs), and we report the exact model architectures, training configurations, and evaluation metrics. The code for algorithm computation open-sourced. Together, these descriptions ensure that our results can be independently verified and extended by the community.
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Appendix
A USAGE OF LARGE LANGUAGE MODELS (LLMS)
In accordance with the ICLR 2026 policy on the use of LLMs, disclose that large language models were employed solely as general-purpose writing assistants. Specifically, LLMs were only used to improve grammar, clarity, and style of exposition. All of the content assisted with LLMs is finally verified by authors.
B EXAMPLES AND ANALYSIS OF ZERO-MEAN INVARIANT DATA
We aim to show that for data modalities satisfying zero-mean invariance, the operation of zero- mean preserves all structural information relevant to generative modeling. We mainly discuss the Euclidean and non-uniform one-hot cases, which are tested in our experiments.
Euclidean Data. Let {a.} C Rd denote collection of n vectors (e.g.. 3D Euclidean coordi- nates of atoms). Define the sample mean æ -i=1 Di and let क; æ; ã be the centered representation. We claim that related positions are invariant under mean-centering:
T3 2. = (a. 3) (z) = z.j-
(15)
Hence, all geometric properties that depend related positions, such adjacency structures, bond lengths, or conformational shapes, are preserved exactly under centering. Consequently, zero-mean projection retains full information about the relational structure of the data.
One-Hot Categorical Vectors. Let h4 E 1) denote one-hot encoded vector satisfying (h,) 1. and let h - h, be the sample mean across batch of such vectors. Define the centered vector h; = h; h. Note that each h E Rd lies in subspace orthogonal the constant vector 1d, since:
d E(hn,, = (h, =1 -11--16
Eh h0 =0
(16)
#=1
J=1
Moreover, the inner product between two centered vectors h2 and h, satisfies:
(T., b,)) hy) (h.,h) (T,, h2) + (h, h),
(17)
from which it follows that pairwise centered dot products retain sufficient information to distinguish between original categorical identities once the category set is not degenerate (e.g., uniform). Since each one-hot vector l is uniquely defined by single active index, subtracting the global mean merely induces a translation within the categorical simplex. The position of the maximal entry in h still identifies the active class long as does not collapse distinct h2 vectors onto the same centered value. Therefore, for any non-uniform categorical data embedded via one-hot encoding, mean-centering preserves the identity of the active component up to affine transformation of the ambient space. As result, zero-mean preprocessing retains the categorical semantics necessary for generative modeling under Euclidean approximation schemes.
C PROOF OF PROPOSITION 3.1
Cumulants. Let X Rd have moment generating function (mgf) Mx Ele" X] and cumu- lant generating function Ky x (u) = log Mx (u), u E Rd, The k-th cumulant tensor is
8 Kx (u)
(c()) (X)) ii....i
1 1.
Tu,, Sus 1==0
In particular (X(() = p := E[X], ((P(( = E = Cov(X), and (k) = 0 for all
G is Gaussian.
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Setup. Let { {aa) |T=o be the forward (noising) trajectory under variance- prrearce- schedule. Fix t () and 8 > t. Then
aztt + Vija, E,
E~ (O,I), E L wt, asft ā./az (0, (18)
aunssianig--add-imeeeedddeee functional (definition). Let D == (Diag(u): o E IR be the diagonal subspace and define the orthogonal projections
II ( == Diaggdiag E),
For weights and wk (k N 3), define for any random vector I2
K
Hx) = 3 |||TT> D (Cov(z)))l + (K223)
Σ wk
(())
(19)
k=3
In following analysis, we will assume k 2 3, then we can omit the indicator function 1 (.). And the conclusion can be easily applied to k = 2.
Lemma C.1 (Variance preserving propagation of moments/cumulants and contraction of H(K)),
Lett t E T") and s > f. and write == aslt E (0, 1). Under equation 18 with Σ, = Cov(x:) and
BEG = || (z:) F for l 223
Ns pt,
Eg a + (1 a) I,
(1 BR, (k: 2 3).
Consequently,
(x,) 3a -aا1 (2x)
+
Σ wk a 122 BR.t.
(20)
and
K
h
H (x,) 3
Io (Et)
+
Σ
wk
a 1/2-1 BR, > 0
0a
2
k=3
whenever || (E+)]lr Ek=s BR, O. Hence, since = Calt decreases strictly in s for a VP ) is strictly decreasing in s unless is already an independent Gaussian (in which case H(K) = 00. 0).
schedule, H(K) (xx)
Proof. First, H4= [ Vauu + -I-aBle = Vap. For the covariance, write I, =
Va vI a and center by the means:
ag 14 va (x) m4) v1 -e. ae.
Independence and [E(] 0 give
Mg E (x.- (,)( - Ms) ) = =a>; + (1- - a) E[se | = aVi + (1 a) I.
Linearity of 1l and I E D yield
1a (2..) 1P (z) + (1 - a) .a.1 (1) = Ml (2x),
hence || (E || F..
For cumulants, independence implies additivity: ((²) = c() + (Y) when X 1
Y (this follows from (x+ = Kx (u) Ky(u)). Homogeneity follows from Kex (1)
log Bie" ex = -xx(cu) and the chain rule: gl
(ex = c
Kx(u)
(()) c((())
Du,, Oui
u= Oun,.. .........
l==0
Because a Gaussian has ((K)(E) 0 for k 2 3,
(1.) = (Vaxz) + (vI- (2xe)
so ([((() (xx,) )l (1/21(((((x,)) F. Plugging these identities into equation 19 gives equation 20 Finally, since B > 0, wk > and BR, 2 0, the displayed derivative is whenever not terms vanish. As a strictly decreases in s for VP, (x.)(x.) strictly decreases in s unless already identically zero
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Lemma C.2 (f-decomposition via prefix sums). For E (0, 1), define
(2(a) == a - a mm) = am/2 (m++1)/2 (3 - m I K 1),
0x(a) = k/2
and m (..3..... K} define the prefix functionals
H = |[ (
Σ wk 10.|| (0:00l
Then (ml(a) > 0 for all m and E (0, 1), and the closed form equation 20 admits
K
K
(x,) = Σ
(a) (m)))))))
with
Σ
(m)(a) = a012 for each j f K}.
m=2
m=
(21)
Proof. For 0 < a <
<a<1, 1, mm = a1 (1)/111)1)))>>>
> 0 for m S K - and (x(a) = x/2 0. To
prove equation 21 expand the right-hand side:
K
K
K
pm (( = (Σ (m((a)
3 BITT (E))lE++
Σ
Σ mi
mm
0m((a) wk (x:))
k=3
Hence suffices to show the tail-sum identities E Σm=20mle mml()) = a and Σm= 0m(a) 2/2 for each k E KK. For k S 1,
Σ m//22 (2+1)/22 K/2
+.... ( (o(k-1)/2_k122)
+ k/2 k/2
a telescoping sum; the case k K immediate. The identity for = 2 is the same computation
with k = 2. Substituting these tail-sums into the expansion recovers equation 20
Lemma C.3 (Order preservation under prefix dominance). Let A, B be two classes of data set at time t. Assume the prefix dominance
H((m)) (x+)) I (x() for all m = 2,3...... K,
(22)
with at least one strict inequality. Then, for every > (equivalently, every a E (0)
K
K
(x
Σ
(c)
(mm) (x
<
mm H (x - = H (k) (z
(xx))
m==
m=2
and the inequality is strict because all mm) for (0,
Proof. By Lemma C.2 H(K) EK 6m (a) ( with 6, (a) > 0. Applying equa- tion 22 termwise gives (x())(()) I H((K)(x (xx)) B) Strictness follows because at least one index m (x=)))
satisfies H(m") (xx)) < ((m") (xx and 0n. (a) > O, hence the weighted sum is strictly smaller. O
Lemma C.1 shows that for each initialization, 8 + H(K) (xx) is strictly decreasing (unless already at an independent Gaussian). Lemma C33 states that if A is prefi--dminant over B at time t, then ((K) (x/) (x(0(()))) for every t. Therefore, for any threshold 0. the hitting times
Tx(E) == inf(s > t: (xx)) <
satisfy (e(e) < TT(),, which formalizes that under VP the speed to gain sufficient Gaussianity for A is faster than for B whenever A starts closer to Gaussian in the sense of equation 22
Moreover, if there exist nondecreasing functions Pdeps Pks (0) 0 such that for every >
o0) - [0. oc) with (O)
Dep(æ.. S Paopl (H( (x.),,
D(x.) S exr( (F(())()))))
(23)
then, for any tolerances Edep, EDS 0,
T == inf t: Dep(x S Edep} < T
T := <nffs>>t: D(z,)) s <pS} < T
and hence T max (TA, TA
Ty. with strict inequality if equation 22 is strict for some
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D GAUSSIANITY COMPARISON BETWEEN IMAGE AND MOLECULAR DATA DISTRIBUTION
Table 5: Gaussianity comparison between QM9 molecular data and CIFAR-10 image data. All values are reported as mean + variance across 10000 samples, which represent the first 10000 molecules of QM9 training dataset and 10000 images in CIFAR-10 test dataset.
Dataset QM9
KS Distance
MI (total)
2.77 x 10-1 + 4.26 x 10
1.66 10 + 1.10 X 10-3
CIFAR-10
5.33 x 10-1 + 1.04 X 10 8.45 x 10-! + 1.84 x 10-1
Experimental Setup. We compute two Gaussianity indicators the raw data distribution of QM9
molecules and CIFAR-10 (Krizhevsky et al. 2009) images: (1) the Kolmogorov-S value, which measures the goodness-of of fit to Gaussian reference (N(0, ū I) for QM9 and (0,
(KS) p-
for CIFAR- 10); and (2) Mhora MMlaae which captures the remaining statistical dependency structure that deviates from Gaussianity. Specifically, MI represents the mutual information(MI) across features, e.g., the atomic positions and types of molecule. The Mlcomp is the MI across components, e.g., different atoms of molecule. All datasets first normalized before compari- son, the molecular normalization follows (Hoogeboom et al. 2022), while the image normalization follows (Ho et al. 2020).
Analysis. As shown in Table 5 both distributional and dependency- metrics indicate that molecular data are substantially closer to Gaussian at initialization than image data. In terms of marginal distributional similarity, QM9 exhibits significantly smaller Kolmogorov-Smimon (KS) distance to the Gaussian reference compared to CIFAR-10, implying that its empirical marginals deviate less from Gaussianity. From the perspective of dependency structure, the total mutual in- formation (MI), defined as the average of feature-wise and sample-wise MI, is markedly lower for contrast, CIFAR-10 retains
QM9, reflecting weaker statistical dependence among dimensions. pronounced non-Gaussian characteristics, manifested by both a larger KS distance and substan- tially higher total MI. Together, these results confirm that molecular data distributions are inherently closer to Gaussian than natural image data, which explains why molecular trajectories attain suffi- cient Gaussianity at earlier noising steps.
E PSEUDO CODE OF ESTIMATION T IN GAGA
Algorithm 1: Pseudo Code of Estimation T" in GAGA
Input: Forward noised trajectory (xx (x=)seo,, which is discretized T steps; data dependency threshold Edep: distributional similarity threshold eps; statistical test frequency
À E (0, 1].
Output: Estimated GA timestep T*
Initialize: TiD + 0c, TDs + oc, counter cnt +- 0.
for t = XT, 2XT, 2x,. do
Compute dependency score Dep(x.). Compute KS distance D, between x; and (0,,, iz7).
Dep(z,) Edep and D. S eps then Increase cnt + cnt + if cnt then Record 7 <- and break.
else
L Reset cnt + 0.
// Two consecutive passes to ensure the statistical reliability.
return T
19

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Published as a conference paper at ICLR 2026
F GAUSSIANITY TEST DETAILS
F.I DATA DEPENDENCY TEST
By treating 2y sample from intractable noised data distribution, estimate the empiri- cal mutual information (MI) across both the sample-wise and feature-wise dimensions of the data tensor. MI quantifies the degree of statistical dependency between random variables by measuring the divergence between their joint distribution and the product of their marginals. For two random vectors X and Y, the mutual information defined as:
((X;Y) f px,y px,y (z. y) log PX. (z. v) (x py py
dr dy,
(24)
where (z, y) denotes the joint probability density, and px (z), PY (y) are the marginal densities of X and Y, respectively.
Given data matrix representation of RNxx we estimate the dependency from two comple- mentary perspectives: feature-wise and component-wise. In molecular data, N is the number of atoms, and d includes both 3D coordinates and atom-type features. Feature-wise MI evaluates the dependency within each atom's feature vector (e.g., correlations among its coordinate and type en- tries), while component- wiss MI evaluates the dependency across atoms for each feature dimension (e.g., correlations among all atoms' 3D Euclidean coordinates). Formally, let
Yeat = {-; E Rd i== N,, Keomp = (x(ب) RN =j==,...... I...,dd, d].
(25)
denote the sets of feature-wise and component-wiss slices, respectively. Then the empirical MI scores are given by
MI
I(x),
MIcamp
Σ [(x).
(26)
[KKoupp
2
where I (x) denotes the estimated mutual information of the given vector across its components.
The data dependency is then estimated by Dep(æ:) MI #Tea+MM.. Ml As f increases, these MI statistics
decay toward zero, indicating diminishing dependency and the emergence of approximate indepen- dence in Ør our experiments, we set the MI threshold Edep = 0.1.
F.2 DISTRIBUTIONAL SIMILARITY VIA KS-TEST
To evaluate whether the noised data 3y has become sufficiently similar to the Gaussian distribution (0, 6,7), we perform statistical testing based on the Kolmogorov-Smiret (KS) criterion. At each test timestep f, we apply the one-sample KS test dimension-wise to the components of 2, after zero- centering, treating each variable as an independent sample drawn from the empirical distribution. Specifically, for each dimension function (CDF) (2) and compare it against the theoretical CDF &; (I) of a univariate normal distribution with zero mean and variance i, derived analytically from the forward noise schedule.
(1....,d d), we compute the empirical cumulative distribution
The test statistic is defined as:
DLJ sup (x) P2. (x)]
(27)
For each coordinate j. we test Ho ;, ( NO, 8x) against H1 :, + (0 0.). Under Ho- with sample size n, the scaled statistic Vn, Dr.j converges to the Kolmogorov distribution with exp(-2k=>2 yielding the 5% critical threshold Dr.j > co.05/ /nn (with (-1)* (-1)*=1
CDF 1 - 2 000
c0.05 Rs 1.36 asymptotically). We declare that timestep satisfies the Gaussianity criterion if at least 95% of coordinates fail to reject Ho, i.e., is statistically indistinguishable from the reference
(O, E,4) at 95% confidence level.
Computational Complexity and Model-Agnostic Property. Both the data dependency and dis- tributional similarity tests in our GAGA scale on average as O(N log N) with respect to the number of samples N. Since these tests are performed only once prior to training, their cost is negligible compared to the iterative computation required for model training over hundreds of diffusion steps. Moreover, the evaluation is fully model- modellaggostic:: On each tested timestep. the injected noise is sam- pled independently; once the Gaussianity timestep T identified, it can be fixed and reused across different generative backbones without any need for re-computation.
20

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Published as a conference paper at ICLR 2026
G EXPERIMENTAL SETTINGS
G.1 BACKBONE MODEL
In our experiments, all molecular generation baselines utilize the Equivariant Graph Neural Network (EGNN) (Satorras et al. 2021) as the backbone architecture for generative processing. EGNN op- erates on graphs embedded in Euclidean space and are designed be equivariant under rigid-body transformations from the special Euclidean group SE(3), including rotations and translations. This property ensures that molecular outputs transform consistently with the input geometry, preserving critical physical symmetries.
Formally, consider molecule represented as a fully connected graph with N nodes, where each node i has coordinates æ; E TR3 and associated atom features h; E Rd. At each EGNN layer, node features and positions are updated through message-panee pasaing operations:
mij 0e (h; h2, | 2;1l²),
h = on
his Σ aiy
(28)
jti
2/ 02
3 x; xy
|| + On (hi, h 1|a0 By a
where des OR, and ohe are learnable functions (typically MLPs), and 0ij is optional attention or reweighting term. mij is the passed message. The update rule guarantees that output features are equivariant with respect to SE(3) transformations. This equivariant structure is critical for molec- ular generative tasks, as the physical properties of molecules are invariant to coordinate shifts and rotations.
G.2 IMPLEMENTATION DETAILS
For all baseline models, we follow the official open-sourced codebases and retain their default hy- perparameters unless otherwise specified. Gaussian Approximation is applied after the truncation step T", as estimated via our KS and MI-based Gaussiamity evaluation. Moreover, to investigate the effect of GAGA on training, we retrain EDM and GeoLDM using truncated generation trajectories, i.e., the forward noise steps are limited to 7* while reducing the training epoch identical in pro- portion to truncations of trajectories. For instance, the original EDM (Hoogeboom et al. 2022) configuration on the QM9 dataset, the model is trained with a forward trajectory of 1000 steps over 3000 epochs. After applying GAGA, the estimated which reduces the maximum noising level used during training. To maintain comparable training sufficiency under this truncated trajectory, we scale the training epochs proportionally to the length 350
auuusian-approximattit
timestep 7 550,
of the preserved trajectory, i.e., from 3000 epochs to 3000 x 1000 1650 epochs. Consequently, the sampling will also start from 550 steps with the analytically derived Gaussian. This adjustment en- sures that the model receives equivalent amount of optimization per effective timestep, while still benefiting from a substantially shorter training trajectory. Notably, even under this reduced training budget and without modifying any architectural hyperparameter settings of the baseline models, GAGA consistently improves both generation quality and efficiency over the original versions.
All molecular generation evaluation metrics are computed on 10,000 generated molecules using RDKit (Landrum et al. 2016). Validity and atom stability are defined by valency correctness, and uniquemess computed as the percentage of distinct canonical SMILES. Sampling time is measured as the average GPU seconds to generate one molecule, while training time reflects total GPU days until the last pre-defined epochs the official repositories.
All experiments are conducted on computing cluster equipped with NVIDIA RTX 3090 GPUs, each with 24 GB memory. Training is parallelized across GPUs using PyTorch DDP framework, while inference experiments are executed on a single GPU for fair comparison of sampling speed. The CPUs are Intel(R) Core(TM) 19- 12900KF. Unless otherwise specified, we report sampling time the average GPU seconds per generated sample, and training time in GPU days until the epochs from the baselines' official repositories. All baseline implementations use their official code, pre-trained weights (if available) and hyperparameters ensure comparability.
21


Reviews:
[{"review_id": "lAnP91qZMB", "reviewer": "Reviewer_1nJS", "summary": "This paper proposes an interesting approach for accelerating the training and sampling procedures of 3D molecular diffusion and flow-based models using Gaussian Approximation. The key idea is that molecular data becomes \"sufficiently Gaussian\" early on in the forward noising process; therefore, the trajectory can be truncated and replaced by closed-form Gaussian distributions. Experiments yield promising results in terms of improved sampling and training efficiency, as well as enhanced generation quality, on two 3D molecular datasets.", "questions": "1. Is the term \"Gaussian Probability Path based Generative Models (GPPGMs)\" known in the literature? If yes, please cite the corresponding papers. If not, please somehow clarify this or use more well-known terms.\n2. Figure 1 is not clear. What is \"complex\", and what is the difference between the two backward passes? Please clarify the figure and the caption.\n3. In Equation 2, $\\alpha_{t-\\Delta t|t}$ is not defined.\n4. In Equation 7, why is $\\tilde v_t$ chosen in this way? Please clearly explain this. \n5. In Equation 10, shouldn't $m>=3$?\n6. In line 225, it is stated that \"sparse molecular coordinates around equilibrium are closer to Gaussian\". Please elaborate and give concrete qualitative or quantitative arguments.\n7. When estimating all the quantities required for computing $T^*$, e.g., the variance of the Gaussian, the dependency evaluator, and the distributional similarity, do you sample multiple noise vectors for each data sample and each noise level?\n8. What are the values of $\\epsilon_{dep}$ and $\\epsilon_{DS}$ and how are they chosen?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "* The idea of assessing the Gaussianity of the data during the noising process to accelerate the training and sampling is novel and sound. Additionally, leveraging this specifically for molecular data is interesting (although some claims in this regard require further clarification, as noted in the weaknesses).\n* The paper is well-written and clearly explained.\n* The experiments covering two common datasets and a few baselines are rather extensive and show promising results.", "weaknesses": "* The main assumption in this paper is that molecular data converges to Gaussianity faster than other modalities, e.g., images. However, this assumption is neither theoretically justified nor empirically verified. Proposition 3.1 requires that the initial distribution is more Gaussian; however, can this be demonstrated? Additionally, based on Figure 2, the authors claim that image data retains recognizable features for more steps than molecular data; however, this can be misleading, as images are much more familiar to the human eye than molecules. Quantitative or qualitative results are needed to verify this claim. For instance, it would be interesting to plot the proposed dependency evaluator $Dep(x_t)$ and the distributinal similarity $D_t$ over different timesteps during the noising process for an image dataset and a molecular dataset (of course, they should be adequately scaled to accommodate the different data scales and ranges).\n* While the paper does a good job at explaining how $T^* $ is obtained, it is missing some details on how the training and sampling procedure is adapted. It seems that the segment $[T^*, T]$ is simply truncated; however, the paper should formally clarify this. For example, what are the exact modifications to the training and sampling algorithms from EDM [1]?\n* All reported experimental results lack standard deviations across different runs, which makes it hard to assess the statistical significance of the claimed results. Also, the code is not provided.\n* It would be a great addition to the paper if the proposed Gaussianity test is used in other settings beyond accelerating training and sampling procedures. In some cases, efficiency is not the primary concern, and the user may be willing to sacrifice efficiency for improved generation quality. I believe the proposed approach can be leveraged in this setting. For example, given a pre-specified noise schedule, find the total time steps $T$ such that the optimal $T^* =1000$ or use the proposed Gaussianity test to guide the design of noising schedules such that $T^*=T$. This setting would enable comparison with baselines for the same computational budget, potentially yielding improved generation results because of better allocation of time steps.\n\nI would be willing to increase my score if the authors address these points convincingly.\n\n**References**\n\n[1] Hoogeboom, Emiel, et al. \"Equivariant diffusion for molecule generation in 3d.\" International conference on machine learning. PMLR, 2022.", "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": "BG7Vt3Dp9f", "reviewer": "Reviewer_GRXx", "summary": "This paper proposes an algorithm to calibrate the noise schedule for diffusion model training and sampling. The central idea is to identify a minimal log signal-to-noise ratio (log-SNR), $\\lambda_\\min$ (in the sense of [1]), at which the noised latent variable attains sufficient Gaussianity. The schedule is then truncated at this point to avoid unnecessary and potentially harmful levels of noising. The effectiveness of this calibration method is demonstrated on several diffusion models for molecule generation.\n\n[1] Kingma, Diederik, and Ruiqi Gao. \"Understanding diffusion objectives as the elbo with simple data augmentation.\" Advances in Neural Information Processing Systems 36 (2023): 65484-65516.", "questions": "1. Could the authors please provide a formal, mathematical definition for the term \"data identity\" as it is used in the context of this paper?\n2. The paper mentions benefits for training. Could the authors clarify precisely how the proposed calibration method reduces training time?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "- Clear Motivation: The paper is well-motivated, addressing the common and practical problem of inefficiently wide noise schedules in diffusion models.\n- Strong Empirical Support: A key strength is the empirical demonstration that baseline models often operate on an unnecessarily broad SNR range. The results convincingly support the claim that truncating this range to an \"effective\" one can be done without degrading model performance.\n- Practical Significance: The proposed method offers a practical tool for scheduler calibration. A significant benefit is the potential to accelerate inference by reducing the number of sampling steps, even when applied post-hoc to models trained with a standard, non-calibrated scheduler.", "weaknesses": "- Insufficient Positioning: A significant weakness is the lack of thorough positioning against the extensive existing literature on noise schedulers, SNR analysis, and related diffusion model theory (e.g., [1,2,3]). This omission causes the work to feel disconnected from established formalism in the area. Consequently, the development of the method appears somewhat ad-hoc rather than being rigorously derived from first principles.\n- Clarity and Readability: The paper's clarity could be improved. A considerable portion of the text is dedicated to standard, boilerplate descriptions of diffusion models, which detracts from the space available to elaborate on the novel contributions.\n- Undefined Key Terminology: A central term, \"data identity,\" is used repeatedly throughout the manuscript but is never formally defined. This ambiguity leaves a core concept of the paper open to interpretation and hinders the reader's ability to fully grasp the method.\n- Unconventional Presentation: In a minor editorial point, the results tables use boldface to highlight the proposed method's results rather than the conventional practice of bolding the best-performing method for a given metric. This unconventional choice makes direct performance comparisons less intuitive.\n\n[1] Kingma, Diederik, and Ruiqi Gao. \"Understanding diffusion objectives as the elbo with simple data augmentation.\" Advances in Neural Information Processing Systems 36 (2023): 65484-65516.\n\n[2] Falck, Fabian, et al. \"A Fourier Space Perspective on Diffusion Models.\" arXiv preprint arXiv:2505.11278 (2025).\n\n[3] Gao, Ruiqi, et al. \"Diffusion models and gaussian flow matching: Two sides of the same coin.\" The Fourth Blogpost Track at ICLR 2025. 2025.", "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": "VKv88LMa5B", "reviewer": "Reviewer_FhFk", "summary": "This paper aims to improve the training and sampling efficiency of diffusion models by truncating the diffusion noise-adding process. For zero-mean data, the authors design an empirical method to select diffusion timesteps, which is validated on molecular generation tasks. Experimental results show performance improvements over methods like EDM and GeoLDM.", "questions": "1. Is the zero-mean assumption a major limitation of your method? Could normalization be applied to handle data with non-zero means, regardless of the original data distribution?\n2. What would be the performance if timestep truncation is applied only during sampling (not training)? Would this harm generation quality?\n3. A curious question: for element types alone, is their distribution closer to a Gaussian than that of images?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The proposed method is simple and serves as a \"free lunch\": it eliminates redundant training and sampling overhead in existing models, boosting both efficiency and generation quality.\n\n2. The writing is clear and easy to follow", "weaknesses": "1. The core idea of truncating the noise-adding process is overly straightforward. Similar concepts have already been explored in iDDPM, where the redundancy of original diffusion schedules was identified and new schedules were designed. This method essentially truncates the existing schedule, which introduces two critical issues:\n    - Theoretical inconsistency: Truncation increases the approximation error of the Gaussian prior. The distance between $x_{T^*}$ and the standard Gaussian is larger than that between original $x_T$ and the standard Gaussian.\n    - Practical inconvenience: A new hyperparameter $T^*$ (or two hyperparameters $\\epsilon_{dep}$, $\\epsilon_{DS}$) is introduced , which needs to tune through experiments. \n\nInstead of truncation, a more viable approach could be adjusting the SNR decay rate of the schedule to slow down decay, thereby compressing the time interval during which the distribution approximates a Gaussian.\n\n2. Limited significance of the method: Performance improvements are more pronounced for models with DDPM schedule (e.g., EDM, GeoLDM). Intuitively, the DDPM schedule has a fast SNR decay rate, leading to many high-noise timesteps—hence the truncation method yields more obvious gains. However, for methods that already address schedule redundancy (e.g., GeoBFN, SLDM), the efficiency improvements from this approach are expected to be far more limited.\n\n3. Gaps in research logic and writing: \n    - The paper introduces three metrics to measure the discrepancy between the current distribution and a Gaussian distribution: cumulant tensors (Eq. 8), mutual information (Eq. 12), and cumulative distribution functions (Eq. 14). The theoretical analysis relies on the first metric, while the second and third are used in experiments—creating a logical gap. Is there a need to include so many similar metrics? Would retaining the least computationally expensive one suffice?\n    - Eq. 26 lacks implementation details for \\text{MI}_{\\text{rows}}: Does it compute the mutual information between all pairs of rows and then average the results? Eq. 12: Does it use row-wise or column-wise mutual information? What is the physical meaning of rows and columns? For the matrix representation of $x_t$: If $x_t$ represents coordinates, is the matrix of size $N \\times 3$ (where N is the number of atoms) or $(3N) \\times d_{\\text{feat}}$ (where $d_{\\text{feat}}$ is the feature dimension)? If the latter, does the independence measured by mutual information depend on the network used to extract features?\n    - Unclear visualization in Figure 3: The \"Gaussianity\" in Figure 3 is represented by colors; numerical values would be more concrete. \n\nRef:\nImproved Denoising Diffusion Probabilistic Models.\n\nStraight-Line Diffusion Model for Efficient 3D Molecular Generation.\n\nUnified Generative Modeling of 3D Molecules via Bayesian Flow Networks.", "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)