Title: TADA: Improved Diffusion Sampling with Training-free Augmented DynAmics
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion Models

ODE

SDE

dynamics system
Abstract: Diffusion models have demonstrated exceptional capabilities in generating high-fidelity images but typically suffer from inefficient sampling. 
Many solver designs and noise scheduling strategies have been proposed to dramatically improve sampling speeds. In this paper, we introduce a new sampling method that is up to 186\% faster than the current state of the art solver for comparative FID on ImageNet512. This new sampling method is training-free and uses an ordinary differential equation (ODE) solver.
The key to our method resides in using higher-dimensional initial noise, allowing to produce more detailed samples with less function evaluations from existing pretrained diffusion models. In addition, by design our solver allows to control the level of detail through a simple hyper-parameter at no extra computational cost. We present how our approach leverages momentum dynamics by establishing a fundamental equivalence between momentum diffusion models and conventional diffusion models with respect to their training paradigms. Moreover, we observe the use of higher-dimensional noise naturally exhibits characteristics similar to stochastic differential equations (SDEs). Finally, we demonstrate strong performances on a set of representative pretrained diffusion models, including EDM, EDM2, and Stable-Diffusion 3, which cover models in both pixel and latent spaces, as well as class and text conditional settings. The code is available at https://github.com/apple/ml-tada.

TADA: Improved Diffusion Sampling with Training-free Augmented DynAmics

Abstract

Diffusion models have demonstrated exceptional capabilities in generating highfidelity images but typically suffer from inefficient sampling. Many solver designs and noise scheduling strategies have been proposed to dramatically improve sampling speeds. In this paper, we introduce a new sampling method that is up to 186% faster than the current state of the art solver for comparative FID on ImageNet512. This new sampling method is training-free and uses an ordinary differential equation (ODE) solver. The key to our method resides in using higher-dimensional initial noise, allowing to produce more detailed samples with less function evaluations from existing pretrained diffusion models. In addition, by design our solver allows to control the level of detail through a simple hyper-parameter at no extra computational cost. We present how our approach leverages momentum dynamics by establishing a fundamental equivalence between momentum diffusion models and conventional diffusion models with respect to their training paradigms. Moreover, we observe the use of higher-dimensional noise naturally exhibits characteristics similar to stochastic differential equations (SDEs). Finally, we demonstrate strong performances on a set of representative pretrained diffusion models, including EDM, EDM2, and Stable-Diffusion 3, which cover models in both pixel and latent spaces, as well as class and text conditional settings. The code is available at https://github.com/apple/ml-tada.

1 Introduction

Diffusion Models (DMs; Song et al. [27]; Ho et al. [14]) and Flow Matching [20, 21] are foundational techniques in generative modeling, widely recognized for their impressive scalability and capability to generate high-resolution, high-fidelity images [4, 10, 24]. Both share a common underlying mathematical structure and generate data by iteratively denoising Gaussian noise through a solver.

Sampling from a diffusion model is typically discretized through a multi-step noise schedule. Denoising Diffusion Probabilistic Models sample new noise at every step resulting in a process interpretable as discretized solutions of diffusion Stochastic Differential Equations (SDEs). Meanwhile, Denoising Diffusion Implicit Models only sample initial noise which is reused at every step resulting in a process interpretable as discretized solutions of Ordinary Differential Equations (ODEs). Both ODE and SDE solvers should perform similarly since they merely represent different interpretations of the same Fokker-Planck Partial Differential Equation. In practice, however, ODE solvers often yield lower-fidelity results than SDE solvers as evidenced by the FID scores in [17] when sufficient function evaluations are performed and model capacity is a limiting factor. On the other hand, SDE solvers require finer steps due to the noise injection at every step, which gets amplified through discretization.

Generating high-quality images using DMs often necessitates a high number of steps and therefore a high Number of Function Evaluations (NFEs), substantially increasing computational costs com-

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pared to alternative generative approaches such as generative adversarial networks (GANs) [11]. Consequently, a large research effort has been focused on reducing the NFEs to reach a desired level of sample quality and this is also the focus of our paper. Recent efforts [17, 32] have significantly improved the efficiency of SDE solvers, achieving compelling results with a moderate number of function evaluations. Designing training-free faster solvers by leveraging numerical integration techniques has also received a lot of attention: exponential integrators [15] and multi-step methods [1], and hybrid combinations of these [22, 23, 33, 34]. Remarkably, faster solvers reach near-optimal quality with around 30 NFEs and retain acceptable image generation capabilities even with NFE < 10 .

To further enhance performance, optimization-based solvers [35, 36] have advanced the capabilities of the aforementioned methods, achieving strong results with as few as 5 NFEs. In parallel, trainingbased dynamical distillation approaches [3, 25, 28, 30] have demonstrated effectiveness in reducing the number of function evaluations and can similarly benefit from the incorporation of fast solvers.

In this paper, we propose a new sampling method that is 186% faster than the current state of the art for generating 512x512 ImageNet samples with an FID of 2 . Our method is orthogonal to existing diffusion model sampling techniques, allowing seamless integration with advanced solvers and classifier-free guidance (CFG) schedules simply by transitioning from standard ODE to momentum ODE frameworks. The proposed method is based on momentum diffusion models [6] and on the observation [7] that they can achieve competitive performance with a very low number of function evaluations. Specifically our paper extends the aforementioned works as follows:

We prove the training equivalence between momentum diffusion models [6, 9] and conventional diffusion models, modulo a transformation of the input variables to the neural network. Consequently, the equivalence shows that simple noise or noisy data augmentation together with diffusion loss, including but not limited to momentum diffusion, offers no training benefit . Readers may extend this reasoning to their own settings and verify the algorithms relevant to them.

We then shift our attention to the sampling phase of the momentum system. We propose a new sampling method named Training-free Augmented DynAmics (TADA) that uses higherdimensional input noise and yet enables direct reuse of pretrained diffusion models.

Although no benefits are observed in the training phase in theory, we find that our proposed method, rooted in the momentum system, inherits advantageous SDE properties while employing ODE solvers, enabling both diverse generation and accelerated sampling through momentum dynamics.

We illustrate the superior performance of our approach through extensive experimentation.

2 Preliminary

First, we cover conventional diffusion models and flow matching, we then follow up by introducing momentum diffusion models and we finish this section with exponential integrators. In the rest of this paper, we follow the flow matching literature convention for the direction of distribution transport, e.g. from the prior distribution at time t = 0 to the data distribution at time t = 1 .

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2.1 Diffusion Models and Flow Matching

In Diffusion Models (DM) and Flow Matching (FM), intermediate states x t ∈ R d are sampled from a tractable transition probability conditioned on an initial data point x 1 ∼ p data during training:

The coefficients µ t ∈ R and σ t ∈ R differ depending on the specific model parameterization: mean preserving, variance preserving or variance exploding. Regardless of these differences, the training objectives across various methods remain identical.

For the sake of clarity, we illustrate parameterizing the noise prediction model with a neural network with learnable weights θ . In this case the objective function can be expressed as:

Alternatively, using the linear relationship given by eq.1, we could just as easily parameterize the data prediction model as x θ ( x t , t ) := ( x t -σ t ϵ θ ( x t , t )) /µ t , or even as a velocity prediction model.

Without loss of generality, we define a force term F θ : R d × [0 , 1] → R d as a linear combination of the learned noise ϵ θ and the state x t , which pushes forward x t from the prior towards the p data from t = 0 to t = 1 . During sampling, we use this force F θ to define the following probabilistic flow:

where a t ∈ R , and b t ∈ R are time-varying coefficients defined by specific parameterization of F θ . For example, for a FM parameterization: F θ ( x t , t ) = ( x θ ( x t , t ) -x t ) / (1 -t ) , a t = 0 and b t = 1 .

2.2 Phase Space Momentum Diffusion Model

Several recent studies have explored a more sophisticated diffusion process defined in the phase space [7, 9]. In this space there are N = 2 noise variables, consequently x t , ϵ ∈ ( R d ) 2 . Owing to the linear

dynamics within this framework-similar to those in conventional diffusion models-the transition probability can also be derived analytically:

where µ t ∈ R 2 and Σ t ∈ R 2 × 2 denote the mean and covariance matrix of the resulting multivariable Gaussian distribution. L t ∈ R 2 × 2 is the noise scaling matrix. The superscript denotes the i -th variable in the multi-variable setting, while the boldface notation x t represents the aggregation of variables, i.e., x t = [ x (0) t , x (1) t ] T in this setting.

Notation : In the rest of this document, we use of the Kronecker product to specify that all the weights are identical along the data dimension d . For example, the expression ( µ t ⊗ I d ) x 1 denotes the stacking of the scaled versions of x 1 by µ t ( n ) , e.g. ( µ t ⊗ I d ) x 1 = [ µ t (0) x 1 , µ t (1) x 1 ] T .

The standard training objective for MDM minimizes the approximation error of ϵ (1) :

Note : It should be observed that such techniques require training since the function ϵ θ takes two data inputs and cannot reuse pre-trained conventional diffusion models where ϵ θ takes only one data input.

Just like for conventional diffusion models, various parameterizations of the learned noise ϵ θ remain viable. In particular, AGM[6] and CLD[9] adopt distinct formulations of F θ , defined as linear combinations of ϵ θ and x t , specifically constructed to guide x (0) t from the prior distribution to the target data distribution. Consequently, sampling reduces to solving a similar probabilistic flow:

where A t ∈ R 2 × 2 and b t ∈ R 2 denote time-dependent coefficients. Specifically, in the MDM framework, the function F θ : ( R d ) 2 × [0 , 1] → R d accepts an aggregated input of two variables of dimension d and outputs a single vector of dimension d . The resulting d -dimensional output is subsequently broadcasted across the N = 2 variables through the coefficient b t , a key distinction from conventional DM. The explicit formulations for A t and b t are detailed in Appendix.F.1.

2.3 Sampling with Exponential Integrators

Once a pretrained diffusion model ϵ θ is obtained, the dynamics specified by eq. 2 or 4 can be readily solved. A variety of advanced training-free fast sampling techniques exist, and many of them rely on exponential integrators [15] which presents as the form of eq.5:

Where Φ( · , · ) is the transition kernel induced by A t . Since the first linear component of the ODE can be analytically integrated via the transition kernel, the primary challenge lies in accurately approximating the second nonlinear integration, such as the neural network parameterized function F θ over discretized time steps s < t .

To improve the accuracy of approximating the nonlinear integral in the second term, advanced ODE solvers are employed as approximators Ψ( · , · , · , · ) in eq.5. Higher-order single-step methods such as Heun's method [12] and multi-step explicit methods like Adams-Bashforth [5] have been widely adopted in prior works [17, 22, 23, 33], often in conjunction with exponential integrators. To further enhance accuracy, implicit schemes such as the Adams-Moulton method [2] have also been introduced [34]. The current state-of-the-art solvers integrate these various techniques, and the specific combinations along with the resulting algorithms are summarized in Appendix.F.3.

This work, however, is specifically focused on designing the ODE itself. Therefore, our approach is entirely orthogonal to existing solver methodologies, allowing it to be seamlessly integrated with any solver type, see fig.1 for explanation and demonstration.

Algorithm 1 TADA sampling

Require: discretized times t i ∈ [ t 0 , t 1 , ...t T ] ; ODE Solver as approximator Ψ( · , · , · , · ) (see eq. 5); Pretrained DM x θ ( · , · ) . Transition function: Φ( t, s ) = exp ∫ t s A τ d τ . Cache Q .

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Westart in Sec. 3.1 with a proof for the training equivalence between momentum diffusion models and conventional diffusion models for any N ≥ 1 . This condition is necessary both for the training-free property of our method as well as for the generalization to arbitrarily large N . We then introduce the proposed method itself in Sec. 3.2. Finally in Sec. 3.3, we study the dynamics of the proposed method and how they tie SDE and ODE formulations.

3.1 Training equivalence between Momentum Diffusion Models and Diffusion Models

In Momentum Diffusion Model case, typically, neural networks are parameterized with N ∈ { 2 , 3 } augmented variables as input. This method naively introduce certain problems. While state-of-the-art diffusion models have developed advanced signal-to-noise ratio (SNR) schedules achieving excellent results [16, 17, 19], defining an appropriate SNR within momentum systems is challenging. Each variable in a momentum system inherently has its own SNR and is coupled via a covariance matrix, complicating the definition and practical usage of SNR. Consequently, this complexity has hindered progress in momentum-based diffusion modeling. Here, we show that the SNR of the momentum diffusion model can be characterized as the optimal SNR achievable through a linear combination of multiple variables, as stated in the following proposition.

Proposition 3.1. The training objective of general Momentum Diffusion Models (MDM) (i.e., eq. 3) can be equivalently reparameterized as:

Moreover, the F θ in eq. 4 can be recovered as a linear combination of x θ and x t (see section.3.2 for details). Here, µ T t Σ -1 t µ t is the effective SNR of ( r T t ⊗ I d ) x t which simply is a weighted linear combination of x t by r t .

This proposition holds significant implications despite its apparent conceptual simplicity: It demonstrates that MDM, even when involving multiple input variables to the neural network, can be trained using a single input constructed as a r t -weighted linear combination of the N input variables. As a consequence, the training objective becomes equivalent to that of a conventional diffusion model, addressing debates about potential advantages in training arising from momentum diffusion or trivial

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variable augmentations. Importantly, this also allows direct reuse of pretrained conventional diffusion models within the MDM framework and therefore the training-free property claimed by our method.

3.2 Momentum Diffusion Sampling Methodology

[6] highlighted that MDM can yield promising results with small numbers of function evaluations (NFE); however, the absolute performance at sufficient NFE lags behind traditional diffusion models, revealing fundamental training issues. Due to the observation in Prop. 3.1, such training issues can be trivially resolved, or, even more easily, we can simply plug in the pretrained diffusion model.

Training-free Augmented DynAmics (TADA)

We now present the sampling procedure after plugging the pretrained diffusion model into the system. Now the input of neural network becomes ( r T t ⊗ I d ) x t ∈ R d instead of x t ∈ ( R d ) N as is the case in vanilla MDM with N = 2 . Since µ T t Σ -1 t µ t is the effective SNR in the momentum system, one can simply map it to the time conditioning used in the pretrained model.

We extend the AGM [6] framework to an arbitrary N -variables augmented space. Similarly to Sec. 2.2, we reparameterize the data estimation x θ to the force term F θ , which drives the dynamics of x (0) t toward x (0) 1 ∼ p data explicitly. The matrices involved have the closed form:

To compute the reweighting term r t , the mean and covariance matrix at time t are required. These quantities can be obtained by analytically solving the coupled dynamics of the mean µ t and covariance Σ t , a standard approach [26] in both conventional DM and MDM. Due to space constraints, the explicit analytical expressions are presented in Appendix F.2.

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3.3 Analysis of Sampling Dynamics

In conventional diffusion models, variations in generative dynamics largely stem from differences in time discretization schemes, as discussed in [17]; for example, Variance Preserving, Variance Exploding, and Flow Matching dynamics primarily differ in their time discretization over SNR during sampling. This naturally raises the question of whether momentum diffusion simply constitutes another form of time discretization. To address this, we conduct a detailed analysis of variable y t which is fed into the neural network in Prop. 3.3.

Proposition 3.3. The dynamics of neural network input y t := ( r T t ⊗ I d ) x t is given by:

For the coefficient of w i t , α t and β t , please refer to Appendix. C. ϵ ( i ) t denotes the estimated Gaussian noise for each variable x i t , induced by x θ . This is analogous to the standard diffusion model but extended to the multi-variable case. Please see Appendix.C for more detail.

Remark 3.4. There are two scenarios in which Prop 3.3 degenerates into a mere different time discretization of the conventional diffusion model, irrespective of the value of N . The first occurs when N = 1 , and the second arises when A t is a diagonal matrix, implying that each variable evolves independently. Further details and proof are provided in Appendix. F .5.

As demonstrated in Prop 3.3, the dynamics of the neural network input y t cannot be expressed solely as a function of y t ; rather, an additional pseudo noise term emerges due to interactions among variables, endowing the system with SDE properties, even when solving the deterministic ODE in Eq. 4. Notably, we empirically find that, the diversity of samples generated from the same prior can be explicitly manipulated by scaling the standard deviation of the final variable x ( N -1) 0 by a factor of k , corresponding to the last diagonal entry of the covariance matrix Σ 0 , as illustrated in Fig. 3.

4 Experiments

In this section, we evaluate the performance TADA in comparison with a range of ODE and SDE solvers based on conventional first-order diffusion probabilistic flows. Specifically, we assess diffusion models such as EDM [17], EDM2 [18], in both pixel and latent spaces on the ImageNet-64 and ImageNet-512 datasets [8]. Additionally, we evaluate the flow matching model Stable Diffusion 3 [10] in the latent space.

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4.1 EDMand EDM2 Experiment

We begin by benchmarking our method against UniPC [34] and DPM-Solver++ [23] on both ImageNet-64 and ImageNet-512, using the EDM [17] and EDM2 [18] frameworks. To ensure fair and competitive evaluation, we conduct comprehensive ablations over all combinations of discretization schemes, solver variants, and solver orders available in the respective codebases, selecting the best-performing configurations for comparison. Full ablation results for the baselines are reported in the Supplementary Material. For our method, TADA , we consistently employ the simplest multistep exponential integrator with third-order solvers, using the same polynomial discretization across all experiments.

We evaluate performance using Fréchet Inception Distance score(FID [13]) for both ImageNet-64 and ImageNet-512, and additionally use Fréchet Distance-DINOv2 (FD-DINOv2 [29]) for ImageNet-512. Our results show that TADA consistently outperforms the baselines across all tested numbers of function evaluations (NFEs) in fig. 5 and fig. 6. We also examine the case with N > 2 , which introduces additional pseudo noise as described in Prop. 3.3. In all setups, we use the same discretization over SNR and vary only the standard deviation of x ( N -1) 0 . This configuration results in improved performance on ImageNet-64, as shown in fig 5. However, no consistent improvement is observed on ImageNet-512, which may be attributed to the high capacity of the neural network and limited exploration of time discretization, thereby diminishing the impact of the additional noise perturbation. For consistency, we therefore report results only for the N = 2 in fig. 6 and the rest of experiments.

Fig 4 presents qualitative results under varying NFE budgets. Unlike conventional ODE solvers, which tend to produce increasingly blurred outputs as NFEs decrease, TADA continues to generate

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plausible images with preserved details. Nevertheless, these images may still deviate from the data distribution p data , as indicated by the larger FID metric. See Appendix. G for more examples.

4.2 SD3 Experiment

For evaluations on Stable Diffusion 3 [10], we exclude DPM-Solver++ from comparison, as UniPC has consistently outperformed it in Section 4.1. Additionally, we observe that the default Flow Matching solver in SD3 achieves performance comparable to UniPC. To enrich our baseline set, we include a recently proposed SDE-based solver, Stochastic Adam (SA) [32]. All baseline configurations follow the recommended settings from the official Diffusers implementation. For evaluation, we adopt HPSv2 [31] as our primary benchmark. Specifically, we generate images for all benchmark prompts, producing 3200 samples per seed, and report the mean and standard deviation over three random seeds, thus, totally 9600 images. To ensure consistency, our method uses the same SNR-based discretization scheme as SD3, which is also the default across all baseline implementations. For the sake of consistency, we employ a second-order multi-step solver, regardless of CFG scale.

Fig. 7 presents the quantitative performance of TADA compared to the baselines. Our proposed method consistently outperforms the baselines across all CFG strengths. However, the performance gap narrows as the CFG increases, suggesting that TADA is particularly effective at lower CFG values, where it achieves more pronounced improvements over existing methods. Fig 8 demonstrates that TADA produces images with more semantically coherent content at 25 NFEs than the baseline, an observation further supported by the quantitative metrics presented in Fig. 7.

5 Conclusions and Limitations

In this paper, we have identified and elucidated the underlying training principle of the momentum diffusion model, demonstrating that it fundamentally aligns with that of conventional diffusion models. This observation enables the direct integration of pretrained diffusion models into momentum-based systems without additional training. Furthermore, we conducted a thorough analysis of the sampling dynamics associated with this approach and discovered that the implicit system dynamics introduce additional pseudo-noise. Empirical evaluations confirmed that this characteristic indeed enhances the sampling quality across various datasets and pretrained models.

However, our approach also has limitations. On the theoretical side, Prop 3.3 does not disentangle the degrees of freedom associated with the pseudo noise dimension N and the weighting coefficients w ( i ) t . As a result, while Fig.5 demonstrates that increasing system stochasticity can lead to improved outcomes, our method still falls short of matching the SDE results reported in [17] with 511 NFEs. This shortfall stems from the limited control over the injected stochasticity due to the naive choice of A t . On the practical side, experiments with SD3 show that as model capacity and CFG strength increase, the performance differences among solvers, dynamical formulations, and between SDE and ODE sampling become increasingly negligible,a trend that also holds for TADA. Conversely, the results underscore TADA's strength in under-parameterized settings, where modest-capacity DM must tackle large scale, high-dimensional data such as those encountered in video generation. Lastly, TADA currently uses a basic exponential integrator. Evaluating further improvement with more advanced solvers, such as UniPC and DPM-Solver++, remains an critical direction for future work.

Broader Impacts: TADA boosts DM efficiency, accelerating the spread of generated content.

References

Kendall Atkinson, Weimin Han, and David E Stewart. Numerical solution of ordinary differential equations . John Wiley & Sons, 2009.

R Bank, RL Graham, J Stoer, and R Varga. Springer series in 8. 1987.

David Berthelot, Arnaud Autef, Jierui Lin, Dian Ang Yap, Shuangfei Zhai, Siyuan Hu, Daniel Zheng, Walter Talbott, and Eric Gu. Tract: Denoising diffusion models with transitive closure time-distillation. arXiv preprint arXiv:2303.04248 , 2023.

James Betker, Gabriel Goh, Li Jing, Tim Brooks, Jianfeng Wang, Linjie Li, Long Ouyang, Juntang Zhuang, Joyce Lee, Yufei Guo, et al. Improving image generation with better captions. Computer Science. https://cdn. openai. com/papers/dall-e-3. pdf , 2(3):8, 2023.

John C Butcher. Numerical Methods for Ordinary Differential Equations . John Wiley & Sons, 2008.

Tianrong Chen, Jiatao Gu, Laurent Dinh, Evangelos A Theodorou, Joshua Susskind, and Shuangfei Zhai. Generative modeling with phase stochastic bridges. arXiv preprint arXiv:2310.07805 , 2023.

Tianrong Chen, Guan-Horng Liu, Molei Tao, and Evangelos A Theodorou. Deep momentum multi-marginal schr \ " odinger bridge. arXiv preprint arXiv:2303.01751 , 2023.

Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition , pages 248-255. Ieee, 2009.

Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically-damped langevin diffusion. arXiv preprint arXiv:2112.07068 , 2021.

Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning , 2024.

Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems , pages 2672-2680, 2014.

Karl Heun et al. Neue methoden zur approximativen integration der differentialgleichungen einer unabhängigen veränderlichen. Z. Math. Phys , 45:23-38, 1900.

Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems , 30, 2017.

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. arXiv preprint arXiv:2006.11239 , 2020.

Marlis Hochbruck and Alexander Ostermann. Exponential integrators. Acta Numerica , 19: 209-286, 2010.

Emiel Hoogeboom, Jonathan Heek, and Tim Salimans. simple diffusion: End-to-end diffusion for high resolution images. In International Conference on Machine Learning , pages 1321313232. PMLR, 2023.

Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models. Advances in Neural Information Processing Systems , 35: 26565-26577, 2022.

Tero Karras, Miika Aittala, Jaakko Lehtinen, Janne Hellsten, Timo Aila, and Samuli Laine. Analyzing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 24174-24184, 2024.

Diederik Kingma and Ruiqi Gao. Understanding diffusion objectives as the elbo with simple data augmentation. Advances in Neural Information Processing Systems , 36:65484-65516, 2023.

Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747 , 2022.

Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003 , 2022.

Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems , 35:5775-5787, 2022.

Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpmsolver++: Fast solver for guided sampling of diffusion probabilistic models. arXiv preprint arXiv:2211.01095 , 2022.

Dustin Podell, Zion English, Kyle Lacey, Andreas Blattmann, Tim Dockhorn, Jonas Müller, Joe Penna, and Robin Rombach. Sdxl: Improving latent diffusion models for high-resolution image synthesis. arXiv preprint arXiv:2307.01952 , 2023.

Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint arXiv:2202.00512 , 2022.

Simo Särkkä and Arno Solin. Applied stochastic differential equations , volume 10. Cambridge University Press, 2019.

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456 , 2020.

Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In International Conference on Machine Learning , pages 32211-32252. PMLR, 2023.

George Stein, Jesse Cresswell, Rasa Hosseinzadeh, Yi Sui, Brendan Ross, Valentin Villecroze, Zhaoyan Liu, Anthony L Caterini, Eric Taylor, and Gabriel Loaiza-Ganem. Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models. Advances in Neural Information Processing Systems , 36:3732-3784, 2023.

Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi. Learning fast samplers for diffusion models by differentiating through sample quality. In International Conference on Learning Representations , 2021.

Xiaoshi Wu, Yiming Hao, Keqiang Sun, Yixiong Chen, Feng Zhu, Rui Zhao, and Hongsheng Li. Human preference score v2: A solid benchmark for evaluating human preferences of text-to-image synthesis. arXiv preprint arXiv:2306.09341 , 2023.

Shuchen Xue, Mingyang Yi, Weijian Luo, Shifeng Zhang, Jiacheng Sun, Zhenguo Li, and ZhiMing Ma. Sa-solver: Stochastic adams solver for fast sampling of diffusion models. Advances in Neural Information Processing Systems , 36:77632-77674, 2023.

Qinsheng Zhang and Yongxin Chen. Fast sampling of diffusion models with exponential integrator. arXiv preprint arXiv:2204.13902 , 2022.

Wenliang Zhao, Lujia Bai, Yongming Rao, Jie Zhou, and Jiwen Lu. Unipc: A unified predictorcorrector framework for fast sampling of diffusion models. Advances in Neural Information Processing Systems , 36:49842-49869, 2023.

Wenliang Zhao, Haolin Wang, Jie Zhou, and Jiwen Lu. Dc-solver: Improving predictor-corrector diffusion sampler via dynamic compensation. In European Conference on Computer Vision , pages 450-466. Springer, 2024.

Kaiwen Zheng, Cheng Lu, Jianfei Chen, and Jun Zhu. Dpm-solverv3: Improved diffusion ode solver with empirical model statistics. Advances in Neural Information Processing Systems , 36: 55502-55542, 2023.

Appendix

A Notations

In this appendix, we follow the notation introduced before. However, in order to reduce the complexity of the derivation, we simplify all the coefficient with Kronecker product. All the variable which sololy depends on the time, will be broadcast by the kroneker product, For example A t := A t ⊗ I d , µ t := µ t ⊗ I d , L v t v = L v t v ⊗ I d etc.

B Proof of Proposition. 3.1

Proof. We first analyze the objective function of momentum diffusion model for N = 2 case, and it can be generalize to larger N . For clarity in dimensional alignment and derivational correctness, we refer the reader to the simplified notation in Section A, which will also be adopted throughout this section.

Following the same spirit, one can derive the case for N variable. See Appendix. J for details.

We know that,

Define and the residual ('noise')

Since the operation is linear, one can transform the x t by

and it is invertible with linear inverse x t = y t µ t + ϵ . Hence the σ -algebras coincide:

For any integrable random variable Z , equal σ -fields imply E [ Z | x t ] = E [ Z | y t , ϵ ] . Taking Z = x 1 yields

due to the fact that ϵ is the independent gaussian, thus

C Proof of Proposition. 3.3

For clarity in dimensional alignment and derivational correctness, we refer the reader to the simplified notation in Section A, which will also be adopted throughout this section.

Proof. The dynamics we considered reads

and again, in this paper, we only consider,

If we expand the system, it basically represent:

(17)

One can easily verify that, when N = 1 , it actually degenerate to flow matching:

One can consider it as the higher augment dimension extension of flow matching model. And the magical part is that, we do not need to retrain model.

For better analysis, we rearange the system:

In our case, the F t function is:

Consider the linear time-varying system:

Since ( 26) is linear and deterministic (apart from the random initial condition), the state remains Gaussian. Its mean and covariance evolve as follows [26].

Mean Dynamics. Let

so that by dividing by x 1 , we obtain

Then

We can write the mean in a factorized form as

Covariance Dynamics. Similarly, the convariance follows dynamics

Recall that, Then we define the scalar quantity of interest as

We wish to compute the derivative of

Using the quotient rule,

Since r T t x t = y t γ t , this becomes

Derivative of r t

Recall that

Differentiating gives

Substitute the known dynamics:

It follows that

Derivative of x t

From ( 26),

Derivative of γ t

Recall

Differentiating, we have:

Thus,

Using ( 30) and ˙ µ t = ˆ A t µ t + ˆ b t , one obtains (after cancellation) the result:

Combining Everything

Substitute the pieces into

Using

Recall that γ t = µ t T Σ t -1 µ t and ˙ γ t = 2 ˆ b T t Σ t -1 µ t . Also note that

Thus, the final expression becomes

The frist term is essentially one kind of linear combination of x t , and Recally that y t = r T x t := µ T t Σ -1 t µ T t Σ -1 t µ t which is another linear combination of x t . Assume that x t ∼ N ( µ t x 1 , Σ t ) , thus, one can derive the relationship between the frist term and y t . Thus, according to Lemma. I.1

Thus, by plugging in the expression, one can get:

D Experiment Details

Here we elaborate more on experiment details.

D.1 EDMand EDM2

For the baselines on EDM and EDM2 codebase, we directly use the code provide in DPM-Solverv3 [36]. For the fair comparision, for all baselines, we controlled σ min = 0 . 002 and σ max = 80 as suggested in the original EDM and EDM2 paper.

For DPM-Solver++, we did abalation search over order ∈ [1 , 2 , 3] , discretization ∈ [ logSNR , time uniform , , edm , time quadratic ] .

For UniPC, we did abalation search over order ∈ [1 , 2 , 3] , discretization ∈ [ logSNR , time uniform , , edm , time quadratic ] ,variant ∈ [ bh1 , bh2 ] .

For all the ablation results, please see the supplementary material.

D.2 Stable Diffusion 3

For stable diffusion 3, we simply plug in the implementation of all the baselines provided in the Diffuser. We use latest HpsV2.1 to evaluate generate dresults.

E Additional Plots

E.1 General N variable dynamics

This section is not referenced in the main paper and will be removed soon; it is retained only for now to keep the appendix numbering aligned with the main paper.

F Detailed Explanations

F.1 Explicit form of A t and b t

Here we demonstrate the A t and b t used in AGM [6] and CLD [9]. Here we abuse the notation and inherent the notation from CLD.

F.2 Explicit form of mean and convariance matrix

Here we first quickly derive how the F derived which is straight-forward. We know x (0) t := x t be the position and define higher derivatives recursively

Where,

The system dynamics form an N th-order chain of integrators driven by a scalar input F ( t, x t ) :

Equivalently, the position satisfies the scalar ODE

Our goal is starting at some time t ∈ [0 , 1) with known state { x ( k ) t } N -1 k =0 , choose F so that the position reaches a prescribed value at t = 1 :

Assume F is held constant over the remaining interval [ t, 1] . Repeated integration yields the degreeN Taylor polynomial about t :

Thus, one can simply solve the F by Rearranging (48) to isolate F :

Thanks to the simple form of F , one can readily write down the mean and covariance of the system. Now we know

By rearraging the dynamics, gives the linear time-varying closed loop

We need first compute the transition matrix induced by ˆ A t and we call it controlled transition matrix. By Solving ˙ Φ = ˆ A t Φ column-wise gives the polynomial matrix

Plugging this Φ( t, 0) into the boxed formulas above supplies µ ( t ) and Σ( t ) explicitly for every order N .

Let µ ( t ) = E [ x t ] . Because ˆ b t x 1 is deterministic,

Define the state-transition matrix Φ( t, τ ) of ˆ A t :

Then the standard variation-of-constants formula gives

If the initial derivatives are i.i.d. N (0 , 1) , then µ 0 = 0 and only the integral term remains. Carrying out the integral (polynomials of τ ) yields

Meanwhile, the propagation of covariance matrix is:

Eq. 51 is a homogeneous Lyapunov ODE whose unique solution is exactly (see Appendix. H for more details):

F.4 Extended Flow Matching

In the framework of flow mathcing, one obtain the velocity by v t = x 1 -x t 1 -t because it is the linear interpolation between data x 1 and prior x 0 . And meanwhile, it happens to be the solution of optimal control problem:

F.3 Previous Fast Solver

For the detailed derivation, please see Sec.C.1 in [6].

For AGM, they consider a momentum system, which reads

The differences is that, AGM consider the injection of stochasticity in the velocity channel. For our case, the F θ derived in Sec. 3.2, is the solution for

and its spirit keeps same as previous formulation, move x (0) t to x 0 1 ∼ p data from t = 0 to t = 1 .

F.5 Degenerate Case of TADA

Here we discuss about the degenerated case of TADA. The reasoning behind it is rather simple. We show the dynamics of y t (eq. 39) again here,

and recall that

Thus, if the first term depends exclusively on y t , the system reduces to the scalar ODE for y t and becomes formally identical to other diffusion-model parameterizations such as VP, VE, or FM. More precisely, in order to degenerate TADA, one only requires

where ˆ b t is defined in eq. 24. This proportionality holds in two scenarios:

When N = 1 , so that µ t and ˆ b t are scalars. In that case, the framework collapses to flow matching-a mere reparameterization of the diffusion model.

When A t is diagonal and its components evolve independently. Then every dimension of µ t and ˆ b t shares the same mean and variance, and proportionality follows directly.

A simple empirical check is to propagate the model from different random initializations using our formulation: it yields identical FID scores after generation, confirming the degeneracy.

G Additional Qualitative Comparision

Please see fig. 10 and fig. 9.

H Solution of the homogeneous Lyapunov ODE

Let Φ( t, τ ) be the state-transition matrix of the (possibly time-varying) coefficient A t :

Throughout we abbreviate Φ( t, 0) ≡ Φ( t ) .

MSD(ours)

UniPC

MSD(ours)

UniPC

1. Candidate solution. Consider

Verification. Differentiate previous equation and use the product rule together with ˙ Φ( t ) = A t Φ( t ) and d dt Φ( t ) ⊤ = Φ( t ) ⊤ A ⊤ t :

Hence Σ( t ) satisfies the differential equation in (Lyap), and Σ(0) = Φ(0)Σ 0 Φ(0) ⊤ = Σ 0 .

Uniqueness. Lyapunov Equation is linear in the matrix variable Σ ; by the Picard-Lindelöf theorem its solution is unique. Therefore (S) is the solution.

I Correlation between two Gaussian Variable

Lemma I.1. Let the random vector

For two fixed column-vectors r , e ∈ R d set

Write the convenient abbreviations

Then

Proof.

Any vector can be decomposed into the component along s and the component orthogonal to s :

Because ϵ ∼ N ( 0 , I d ) and the projector onto g t is orthogonal to the projector onto the complement, g T t ϵ and ϵ ⊥ are independent Gaussian variables.

Insert g T t ϵ = y t -c t to obtain

Then we can plug this into z t :

J General N variable MDM loss

At time t the N -variables are generated by

with known µ t ∈ R N and invertible L t ∈ R N × N . The goal is to predict ε ( N -1) .

By whitening trick, one can isolate ϵ ( N -1) . Let e ⊤ N -1 = [0 , . . . , 0 , 1] be the row vector that selects the last coordinate. Left-multiplying by e ⊤ N -1 L -1 t gives

Define the time-dependent scalars then

A neural network ε θ ( x t , t ) is trained to approximate ε ( N -1) with the standard

Insert eq. 62 and multiply the interior by b t :

where

̸

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work builds on momentum diffusion models, extending them while proving their training equivalence to conventional diffusion models, thus enabling the reuse of pretrained models without retraining.\n\nThe main contributions are\n1. The authors demonstrate that momentum diffusion models (diffusion models with multi variables) can be reparameterized to match conventional diffusion models’ training objectives, allowing pretrained diffusion models to be directly used.\n\n2. This work introduces a reweighting term to make low-dim noise to high-dim (from N= {1,2} to more), increasing sampling diversity and quality without additional training.\n\nExtensive experiments demonstrate that the proposed methods outperforms state-of-the-art ODE solvers (e.g., UniPC, DPM-Solver++) across various evaluation metrics (e.g., FID, HPSv2).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper presents an interesting and solid contribution, a training-free sampling method for diffusion models that leverages momentum dynamics and higher-dimensional noise augmentation. The theoretical foundation is solid, with formal proofs demonstrating the equivalence between momentum diffusion models and conventional diffusion models, enabling direct reuse of pretrained models. The empirical evaluation is thorough, covering multiple datasets, pretrained models, and baselines, and demonstrates clear improvements in sampling speed and diversity with competitive sample quality.\n\n\nWeakness & Questions:\n\n1. I find it somewhat confusing that the authors set the default configuration to N=2 after conducting ablation studies in EDM2. The results in FIG 5 seem to suggest that a larger value for N yields better performance. Can I expect a comparison in larger model, e.g. SD3? \n\n2. In the process of transformation, do all these parameters have closed-form solutions? The author mentions that A_t exists. As for the other terms, such as r and F_\\theta, are they also lossless? \n\n3. The author seems to suggest that TADA performs better when using small models to handle high-dimensional data. However, I am wondering whether the estimation error of F during sampling might become larger in this case.\n\nQUESTIONS:\nsee weakness\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-14"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes TADA (Training-free Augmented DynAmics), a sampling method for diffusion models that uses higher-dimensional state spaces to reduce the number of function evaluations (NFEs) required for generating high-quality samples. The method establishes a theoretical equivalence between momentum diffusion models and conventional diffusion models, enabling the reuse of pretrained models without additional training.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n 1. This paper introduces an interesting augmented sampling method. \n\n  2. The theoretical equivalence (Proposition 3.1) enables direct application to existing pretrained diffusion models, providing immediate practical value to practitioners.\n\n 3. Empirical results demonstrate consistent reduction in NFEs across multiple models (EDM, EDM2, SD3) and datasets (ImageNet-64/512), showing the method's robustness.\n\n 4. The connection between momentum diffusion and pseudo-noise emergence (Proposition 3.3) provides an interesting bridge between ODE and SDE formulations.\n\n 5.The approach can be integrated with existing advanced solvers, making it a valuable addition to the diffusion sampling toolkit.\n\n\nWeaknesses:\n\n1.The \"186% faster\" claim lacks clearly explanation of how it was measured. Was this based on wall-clock time, NFEs, or some other metric?  \n\n2.While Proposition 3.1 shows equivalence, the fundamental reason why augmented dynamics lead to fewer NFEs remains unexplained. Without a mechanistic understanding, it's unclear when and why TADA works.\n\n3.Given that each TADA step requires additional computational overhead (rt computation, N-dimensional state updates, complex force term calculation), how can the method be faster overall? \n\n4.The supplementary material only provides detailed ablation results for DPM-Solver++ and UniPC baselines, but lacks corresponding results for TADA. Without TADA's ablation studies on different solver orders, it's impossible to verify the fairness of comparisons.\n\nQUESTIONS:\n1.The paper claims TADA achieves both SDE-like properties and faster sampling than ODE solvers. Since TADA's \"pseudo-noise\" (Proposition 3.3) is deterministically derived rather than stochastically sampled, what specific mechanism actually drives the FID improvements compared to standard ODE solvers?\n2.What is the memory usage scaling with N? Are there practical limits for large-scale models? Have you tested N > 8?\n3.How does TADA perform on CIFAR-10? Can TADA further improve generation quality like SciRE-Solver and Restart sampling, rather than just reducing NFEs?  \n4. Why the title of this paper in the attached pdf is \"TADA: Improved Diffusion Sampling with\nTraining-free Augmented DynAmics\", and it is different to the title of \"Momentum Sampling Dynamics for Diffusion Model\" given in the submitted information?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a diffusion sampling method based on the formulation of augmented diffusion model. The proposed method is a training-free approach that leverages traditional pre-trained models and utilizes an ODE solver. By introducing momentum dynamics, it achieves both fast sampling and diversity preservation. They demonstrate the training equivalence between the momentum diffusion model and conventional models, and, based on this, propose a training-free sampler that addresses the training instability of the momentum-based model. The method is empirically validated on various pre-trained models, demonstrating its effectiveness.\n\nSTRENGTHS AND WEAKNESSES:\n## [Strengths]\n\n* They demonstrate the equivalence between the momentum diffusion model and conventional diffusion models, and their theoretical analysis appears sound. This shows that the advantages of the momentum diffusion model, which has training instability, can be leveraged in a training-free manner by using models trained with the traditional diffusion loss. This approach is reasonable and novel, especially considering that similar ideas have been observed in previous diffusion samplers (e.g., DDIM).\n\n* Experiments conducted on various backbones and sampler configurations consistently demonstrate strong performance, empirically showing the method's effectiveness.\n\n* Furthermore, since the proposed approach can be applied adaptively to existing diffusion models and samplers, it has considerable potential for real-world applications.\n\n## [Weaknesses]\n\n* In the SD3 experiment, HPSv2 is used as the evaluation metric. It would be beneficial to add results with FID and CLIP scores on the COCO validation dataset, which are provided in the SD3 paper. This would allow for a more detailed comparison of image quality and text alignment with other models.\n\n* Figure 1 introduces various notations, but no definitions are provided for notations like $r$ until the Preliminary section, making it difficult to understand.\n\n* It would be beneficial to align the formulations of Eq. (3) and the MDM training objective in the proposition. Although the appendix shows how the noise estimation formulation in Eq. (3) leads to the x-prediction formulation in the MDM objective, a unified presentation would help reduce confusion.\n\n* The explanation of Proposition 3.1 states that $F_\\theta$ can be recovered as a linear combination of $x_\\theta$ and $x_t$, but the explanation for this appears later in the middle of Section 3.2. It would be helpful to either provide the formula for $F_\\theta$ here or at least indicate where the explanation can be found.\n\n* In Proposition 3.3, $\\epsilon^{(i)}$  is introduced without a definition. Based on the proof in Appendix B, it seems to refer to $\\epsilon_{\\perp}$, but explicitly mentioning this in the main text would help readers recognize that $\\epsilon^{(i)}$ is a pseudo-noise term.\n\n* The readability of the paper could be improved. For example, there are many instances where citations [] appear without a space, and it would be better to use a consistent format when referring to sections, propositions, and figures (e.g., sometimes it is written as 'Appendix.E.4' and other times as 'Appendix E.2' with or without a period after 'Appendix'). Additionally, for Figure 8, the text condition used to generate the images should be provided in the main text to allow verification of semantically coherent content.\n\nQUESTIONS:\nI would appreciate answers to the questions raised in the Weaknesses section above. In particular, it would be helpful to see additional experiments or validation regarding the SD3 experiments.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- The authors propose TADA, a training-free approach for accelerating the sampling process of diffusion models\n- The method relies on augmented dynamics, where an auxiliary variable (momentum) or many is introduced\n- A key contribution is establishing a link between momentum ODEs and conventional diffusion models. A byproduct of this result is training-free method that allows pre-trained diffusion models to be used within the momentum ODE framework\n- The method is validated on state-of-the-art models, including EDM, EMD2, and Stable Diffusion 3\n\nSTRENGTHS AND WEAKNESSES:\n## Strength\n\n- The paper proposes a training-free approach that enables the use of conventional, pre-trained diffusion models within the more efficient momentum diffusion sampling framework, avoiding the need for costly retraining\n\n\n## Weaknesses\n\n**Theoretical and Methodological Concerns**\n\n- The result suggesting that one can synthesize $E[x_1 | \\mathbf{x}_t]$ from a linear combination of the position and momentum states is concerning. If this is true, it implies that the two state equations are redundant, which would undermine the motivation for introducing an auxiliary variable in the first place.\n- The proof presented in Appendix A contains shape mismatches that invalidate the argument and make it difficult to verify\n    - In Line 716,  $\\mathbf{x}_t \\in \\mathbb{R}^{2d}$ is drawn from a Gaussian with covariance $\\Sigma_t \\in \\mathbb{R}^{2 \\times 2}$, hence incompatible shape.\n    - Similarly, in Line 717, $r_t \\in \\mathbb{R}^2$ cannot be multiplied by a vector in $\\mathbb{R}^{2d}$\n- The statement in Lines 30-33 is overly strong. It is well-established that for a low number of function evaluations (NFE), ODE-based solvers like DDIM often outperform SDE-based samplers; see [1] Table 1.\n- The caption for Figure 2 is concerning, as it suggests the authors are simulating a deterministic ODE yet obtaining stochastic samples. This requires clarification.\n\n**Typos and Broken Links:**\n\n- In Line 155, the reference to \"Prop 3.1\" does not correctly link to Proposition 3.1 in the main paper.\n- In Appendix Line 852, \"Guassian\" should be corrected to \"Gaussian\"\n\n\n---\n\n.. [1] Song, Jiaming, Chenlin Meng, and Stefano Ermon. \"Denoising diffusion implicit models.\" arXiv preprint arXiv:2010.02502 (2020).\n\nQUESTIONS:\n- After reading the appendix, it is still unclear what the exact expressions for $\\mathcal{r}_t, \\mathcal{\\mu}_t$, and $\\Sigma_t$ would be when applying TADA to the conventional DDPM use case [1]. Could the authors provide these?\n- In Figure 2, what is the purpose of plotting the difference between the TADA and EDM trajectories? The takeaway from this comparison is not clear.\n- Regarding the evaluation in Figures 5 and 6, the choice of NFE for the Heun solver baseline (represented by a horizontal dashed line) seems arbitrary. It gives the impression of being a lower bound, but the motivation for this specific choice, especially when TADA performs below it in Figure 6, is not explained.\n\n\n--- \n\n.. [1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. \"Denoising diffusion probabilistic models.\" Advances in neural information processing systems 33 (2020): 6840-6851.\n\nLIMITATIONS:\n.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}