TADA: Improved Diffusion Sampling with Training-free Augmented DynAmics
Authors: Tianrong Chen
Huangjie Zheng
David Berthelot
Jiatao Gu
Joshua M. Susskind
Shuangfei Zhai
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Models
ODE
SDE
dynamics system
Primary area: deep_learning

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
Tianrong Chen, Huangjie Zheng, David Berthelot, Jiatao Gu, Josh Susskind, Shuangfei Zhai
Apple
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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 dramatically improve sam- pling 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-fiee free and uses an ordinary differential equa- tion (ODE) solver. The key to our method resides in using higher- ligher-dimensionll initial noise, allowing to produce more detailed samples with less function eval- uations from existing pretrained diffusion models. In addition, by design our solver allows to control the level of detail through simple hyper-parameter at no extra computational cost. We present how our approach leverages momen- tum 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 nigher-dimensional noise naturally exhibits characteristics similar to stochastic differential equations (SDEs). Finally, demonstrate strong performances models, including EDM, EDM2, and Stable-Diffusion 3, which cover models both pixel and latent spaces, well as class and text conditional settings. The code is available at https//githuh..co/app/////
of representative pretrained diffusion
Introduction
Diffusion Models (DMs; Song et al. [27]; Ho et al. [14) and Flow Matching 21 are foundational techniques in generative modeling, widely recognized for their impressive scalability and capability to generate high-resolution, high-fidelity images TO 24 Both share a common underlying mathematical structure and generate data by iteratively denoising Gaussian noise through solver.
Sampling from a diffuusin model is typically discretized through multi-step noise schedule. Denois- ing Diffusion Probabilistic Models sample new noise every step resulting in 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 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 evidenced by the FID scores in 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 every step, which gets amplified through discretization.
17 when sufficient
Generating high-quality images using DMs often necessitates high number of steps and therefore a high Number of Function Evaluations (NFEs), substantially increasing computational costs com-
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(0)
xo(r'x,
X (0)
zdbig f))
x/
Solver
x
x(N-1)
Solver
(V-1)
1/s1
Figure 1: Here we show the distinctions between conventional diffusion models (left) and momentum diffusion models (right) during sampling. Leveraging Prop 3.1 we demonstrate that pretrained diffusion models with zo-prediction ze(;;)) can be directly applied to propagate the momentum
system with multiple varible (x} (x),)))0=== W- Moreover, the choice of numerical solver remains flexible. See Sec 3.2 for details.
pared to alternative generative approaches such generative adversarial networks (GANs) 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 func- tion evaluations. Designing training-free faster solvers by leveraging numerical integration techniques has also received lot of attention: ex exponential integrators (15] and multi-step methods [. and hybrid combinations of these 22
Remarkably, faster solvers reach near-optimal quality with around 30 NFEs and retain acceptable image generation capabilities even with NFE < 10.
23
33 34
To further enhance performance, optimization-baser solvers 35 36 have advanced the capabilities of the aforementioned methods, achieving strong results with as few as NFEs. In parallel, training- based dynamical distillation approaches B 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 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 77 that they can achieve competitive performance with a very low number function evaluations. Specifically our paper extends the aforementioned works as follows:
We prove the training equivalence between momentum diffusion models 66 9 and conven- tional diffusion models, modulo 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 momentum diffusion, offers no training benefit. Readers may extend this reasoning to their own settings and verify the algorithms relevant to them.
2. We then shift our attention to the sampling phase of the momentum system. We propose new sampling method named Training-frr free Augmented DynAmics (TADA) that uses higher- dimensional 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.
4. 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 to the data distribution at time t 1.

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Pixel Trai. Dif.. between TADA(ours) and EDM-ODE
Pixel Value Differences
EDM2 ODE
TADA ODE (ours)
Traj.
Pixel Value Differences
8
EDM2 SDE
10
101
Effective Noise Scale
Figure 2: Demonstration of differences of TADA and probablistic ODE in terms of trajectories. We apply the same pretrained model with momentum system with same discretization terms of SNR and the same initial prior samples. TADA generate SDE-like property, such generate different samples from same initial condition, but the system keeps deterministic ODE which can be solved more efficiently. See Prop 3.3 for more detail.
2.1 Diffusion Models and Flow Matching
In Diffusion Models (DM) and Flow Matching (FM), intermediate states Ie E R.d are sampled from tractable transition probability conditioned on an initial data point Iı Pdata during training:
Iz 1 + OzE, ~~00 Ia).
The coefficients μr IR and o, 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 0. In this case the objective function can be expressed as:
min LDM (0) Fer,, llleelllo....... ell2.
Alternatively, using the linear relationship given by eq we could just as easily parameterize the data prediction model as xp (2,,, t) == (z, (I,, t)) Nr. even as velocity prediction model.
Without loss of generality, we define a force term F6 _ x [0, 1] + Rd as a linear combination of the learned noise εœ and the state Ht, which pushes forward 2, from the prior towards the Pdata from t = = 1. During sampling, we use this force Fg to define the following probabilistic flow:
dx; = a,s, + beFo, (), 2o N00
(2)
di
where E R, and b, E R are time-varying coefficients defined by specific parameterization of Fa. For example, for FM parameterization: F2 1) (xa (x,, xx)/(1 t), at 0 and b, 1.
2.2 Phase Space Momentum Diffusion Model
Several recent studies have explored more sophisticated diffusion process defined in the phase space 7 9.. In this space there N 2 noise variables, consequently X,, ) Owing to the linear

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dynamics within this framework- similar to those in conventional diffusion models the transition probability can also be derived analytically:
X, - (ee & Ia)x1 + (L, @ Ia)e, L,L, = Ex, E c: = c00)
( ~ NO0 I2a);
where 14, R2 and Σ, E R2x2 denote the mean and covariance matrix of the resulting multi- variable Gaussian distribution. L, E IR2x2 is the noise scaling matrix. The superscript denotes the i-th variable in the of variables, i.e., X = [ xit 1 this setting.
setting, while the boldface notation X represents the aggregation
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 (14 @ Ig)ar denotes
Notation:
the stacking of the scaled versions of x by (1), e.g. (mr @ La)æı [u, 21, feg (2) x1]"
The standard training objective for MDM minimizes the approximation error of (1))
min MDDM (0) = Ex, ee. (x,, --0-((0 10 with co (RR))² x [0, + Rd
(3)
Note: It should be observed that such techniques require training since the function co takes two data inputs and cannot reuse pre-trained conventional diffusion models where co takes only one data input.
Just like for conventional diffusion models, various parameterizations of the learned noise eg remain viable. In particular, 6GM and CLD(2] adopt distinct formulations of Fe, defined linear combinations of co and x,, specifically constructed to guide x, (0) from the prior distribution to the target data distribution. Consequently, sampling reduces to solving similar probabilistic flow:
dx di
= (Ae & Lx)x; + (b, @ (x),), Xo ~ ~(0, Eo).
(4)
where A, E IR2x2 and b, E _² denote time-dependent coefficients. Specifically, in the MDM framework, the function Fg (Rd)²² x [0 R. accepts an aggregated input of two variables of dimension and outputs a single vector of dimension The resulting d-dimensional output is subsequently broadcasted across the N = 2 variables through the coefficient b, key distinction from conventional DM. The explicit formulations for A, and b, are detailed in Appendix F.I
2.3 Sampling with Exponential Integrators
Once a pretrained diffusion model co is obtained, the dynamics specified by eq. or 4 can be readily solved. A variety of advanced training-free free fast sampling techniques exist, and many of them rely exponential integrators [15] which presents as the form of eq 5
Ft (),, + [-
(t, ["
r)b, Fo Fo
Approximator V (s.t.a,,FF))
Where (((.))) is the transition kernel induced by Ac- Since the first linear component of the ODE 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 Fo over discretized time steps
To improve the accuracy of approximating the nonlinear integral in the second term, advanced ODE solvers are employed as approximators v .). in 5 Higher-order single-step methods such as Heun's 's method [2] and multi-step explicit methods like Adams-Bashforth [5] have been widely adopted in prior works [17 22 23 B3], often in conjunction with exponential integrators. To further enhance accuracy, implicit schemes such as the Moonton method 2 have also been introduced (34). The current state-of- offte--f-the-aae 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 for explanation and demonstration.

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Algorithm TADA sampling
Require: discretized times l, [lo (1,,, ODE Solver approximator ( (.)))))) (see eq. 5
Pretrained DM xo(+,)). Transition function: ( s) exp f. AT. Cache Q. draw prior sample
1: Xtg (0, mt,)
2: for = to T
do
3: compute 14; L1, and rt, == Ei, Er,'ac
> obtain the reweighting for N variables
pel, 21, Me,
4: i t 20 ((r), La)xt, ti)
data prediction with pretrained DM
5: Fo t- N f EnO #h (1 14)
n.
b Compute force term, see Sec.
(1 - t:)
Ve, ( hut (11++1,7) by F(x+, T) dr
6:
Approx. nonliear part using existing solver (with Q if solver is multistep)
7: Solver is multistep then cache $
store history b state update
8: X1.41 + (1;-11,4)xx, + Ve:
9:
end for
10:
return zo(( (r & La) Xtx: tT)
> return results with data prediction
Method
We start in Sec. 3.1 with a proof for the training equivalence between momentum diffusion models and conventional diffusion models for any 2 1. 'his condition is necessary both for the training-free property of method well for the generalization to arbitrarily large N. We then introduce the proposed method itself in Sec. 3.2, S.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 imput. This method naively introduce certain problems. While state-of-the-an diffusion models have developed advanced signal-to-noise ratio (SNR) schedules achieving excellent results [16 17 [9]. defining appropriate SNR within momentum systems is challenging. Each variable in 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- diffusion model can be characterized the optimal SNR achievable through linear combination of multiple variables, as stated in the following proposition.
diffusion modeling. Here, we show that the SNR of the momentum
Proposition 3.1. The training objective of general Momentum Diffusion Models (MDM) (i.e., eq. can be equivalently reparameterized as:
(MDM(0) xc Ex, (X1,
, t) x1 112 +
xx(x;,,) t) E[x1[x]]
Σ, pr I. Σ, p
E[2 [x,] E [ (r) & Ix)x, c] [I @ Ixx.] where r, =
Moreover, the F6 in eq. can be recovered as a linear combination of za and x; (see section 3.2 for details). Here, pud E, p4 the effective SNR of (r. & L2)x, which simply a weighted linear combination of x, by rz-
Proof. See Appendix.
B
This proposition holds significant implications despite its apparent conceptual simplicity: It demon- strates that MDM, even when involving multiple input variables to the neural network, can be trained using a single input constructed as re-weighted linear combination of the N input variables. As consequence, the training objective becomes equivalent to that of a conventional diffusion model, addressing debates about potential advantages in training arising from momentum diffusion trivial
5

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k = k= 3 k=5 == 10 k= = 20 k= 25
EDM2-ODE
TADA(ours)
Figure 3: Generated samples under varying prior scales are shown, all initialized with the same initial condition of dynamics in Prop. 3. 3/o == (x) @ Ia)xa c, using an identical time discretization over the SNR and the same pretrained model with 15 NFEs. It can be observed that the diversity of (N-1)
the generated results increases proportionally with the standard deviation the final variable To which is scaled by factor k: Eg diag k,.,,))
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 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 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) @ Ia)x, E Rd instead of x, E (IRd) as is the case in vanilla MDM with = Since μ Σ, f4r 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
framework an arbitrary N-variables augmented space. Similarly to Sec. 2.2 we reparameterize the data estimation 2y to the force term Fo. which drives the dynamics of (0)
It
toward 21
(0)
Pasta explicitly. The matrices involved have the closed form:
0
0
0
0
Fo(x,, N! Io(r Xt. x). - N
N (())
n! -1 1)"
A
b,
(1 t)N
0
OJ
NxN
Remark 3.2. When N = this formulation simplifies precisely to vanilla flow matching. When N 2, it is essentially same the deterministic of AGM [6] but with the added training- property that our method carries. Please see Appendix. for details.
To compute the reweighting term rt, the mean and covariance matrix time are required. These quantities can be obtained by analytically solving the coupled dynamics of the mean μ4, and covariance Σ,, standard approach (26] in both conventional DM and MDM. Due to space constraints, the explicit analytical expressions are presented in Appendix F2

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NFE=5
NFE-7
NFE=15
NFE=21
NFE=5
NFE-7
NFE=15
NFE-21
TADA(ours)
FID=13.41
FID=3.57
FID=1.95
FID=1.955
FID=13.41
FID=3.57
FID=1.95
FID=1.95
UniPC
FID=34.71
FID=9.54
FID-2.23
FID=2.06
FID=34.71
FID=9.54
FID-2.23
FID=2.06
Figure 4: Qualitative comparison with UniPC, varying the NFEs, using the same initial condition and the pretrained EDM2 model. More qualitative comparision can be found in Appendix, G
3.3 Analysis Sampling Dynamics
In conventional diffusion models, variations in generative dynamics largely stem from differences time discretization schemes, as discussed [[7]: 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, conduct detailed analysis of variable y which is fed into the neural network in Prop. 3.3
Proposition 3.3. The dynamics of neural network imput = (r. @ Ia)xe is given by:
N-1
dyt
Σ
wi 10
( (x)
+aih + (er
Pseudo Noise
For the coefficient of wi. ai and Br please refer Appendix. E denotes the estimated Gaussian noise for each variable xi, induced by Ig. This is analogous to the standard diffusion model but extended to the mudti-varialte case. Please see Appendix[C more detail.
Proof. See Appendix.
Remark 3.4. There are two scenarios in which Prop 3.3 degenerates into a mere different time discretization the conventional diffusion model, irrespective of the value of N. The first occurs when N = 1, and the second arises when A, 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/ cannot be expressed solely as a function of yei 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, 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 , (N-1) by a factor of k. corresponding to the last diagonal entry of the covariance matrix Eg, as illustrated in Fig.
4 Experiments
In this section, we evaluate the performance TADA in comparison with range of ODE and SDE solvers based on conventional first-order diffusion probabilistic flows. Specifically, we assess diffusion models such as EDM [7), EDM2 [18] in both pixel and latent spaces on the ImageNer-64 and ImageNet-512 datasets [ Additionally, we evaluate the flow matching model Stable Diffusion 3 [10] in the latent space.
7

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Busciiee Solvr Broetine: UpiPC
2.7
TADA (oun) No2
poichet loegtion Distano (PIDS)
4.5
Frichet Inception Distancs (FTD)
TADA (oun) Nad
TADA (ours)
2.6
(ours)
TADA (ouns) Nad
2,5
3.5
2.4
3.0
2.3
2.1
2.0
2.0
10
15
20
20
30
36
20
30
40
50
60
70
Number of Pinction Fxalusaion (NFF5)
Number Function Evaluation (NFFs)
Figure 5: left: Comparison with baselines on ImageNet-64 using EDM pretrained model. Right: Performance under varying numbers of variables N while keeping the SNR-based time discretization same to the = 2 setting: = 2 is the default configuration reported throughout this paper.
100
5.0
Hovn NER
Hegn NPE
Buseinn DPM-Sulver +
Baseliuer D
Priehet Invertinn Dittano (PID)
Basellnes UnIPC
Dnor2 Peéchct Distance (DINOv2)
Basellon: UniPC
TADA (ours)
90
TADA (ours]
3.5
3.0
2,5
80
50
Number of Function Fosluations (NFF)
Numbur of Pamction Fvaiuations (NFR)
Figure 6: Comparison with baselines on ImageNet-512 with EDM2 pretrained model.
4.1 EDM and EDM2 Experiment
23 on both We begin by benchmarking our method against UniPC 34 and DPM-Solver++
ImageNet-64 and ImageNet-512, using the EDM [17 and EDM2 [[8] frameworks. To ensure fair and competitive evaluation, we conduct comprehensive ablations over all combinations of dis- cretization 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 method, TADA, consistently employ the simplest multi- step exponential integrator with third-order solvers, using the same polynomial discretization across all experiments.
We evaluate performance using Fréchet Inception Distance SFDD 13 for both ImageNet-64 and ImageNet-512, and additionally Fréchet Distance-DINOvs (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. and fig. We also examine the with N 2, which intro- duces 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 xh 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 in fig. 6 and the of experiments.
This configuration results in improved
Fig presents qualitative results under varying NFE budgets. Unlike conventional ODE solvers, which tend to produce increasingly blurred outputs NFEs decrease, TADA continues to generate

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SA-SDE
UniPC-ODE
ToDA(
01.2-
HPSV2 Score
25.0
30.70
51.0
31.
34.0
10:00-
25.2
50.00
NFF R/CRG
NFR w/ CCFF
2.5
NTP w/CFG 3.5
NFE W/CCFF
Figure Comparison with baselines with HpsV2 metrics with SD3 pretrained model.
MSD(ours)
UniPC
NFE=10
NFE=25
NFE=10
NFE=25
Figure 8: Qualitative Comparison with baselines with SD3 model w/o CFG. UniPC delivers no additional gains in image quality, whereas our method continues to improve. This trend is evident both in Fig. and in the samples generated NFE 25.
plausible images with preserved details. Nevertheless, these images may still deviate from the data distribution Pdata, as indicated by the larger FID metric. See Appendix, G more examples.
4.2 SD3 Experiment
For evaluations Stable Diffusion 3 [10 we exclude DPM-Solver++ from comparison, UniPC has consistently outperformed it in Section 4.1
Additionally, we observe that the default Flow Match- ing 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 SD3, which is also the default across all baseline implementations. For the sake consistency, we employ second-order multi-step solver, regardless of CFG scale.
Fig. 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 achieves more pronounced improvements over existing methods. Fig demonsttrates 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.
5 Conclusions and Limitations
In this paper, we have identified and elucidated the underlying training principle of the momentum diffusion model, demonstrating that 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 nseudo-noise. Empirical evaluations confirmed that this characteristic indeed enhances the sampling quality across various datasets and pretrained models.

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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 As result, while Fig demonstrates that increasing system stochasticity can lead to improved outcomes, our method still Talls short of matching the SDE results reported in [7 with 511 NFEs. This shortfall stems from the limited control over the injected stochasticity due to the naive choice of Aç. On the practical side, experiments with SD3 show that 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- paadter-rrameerezeeed settings, where modest-capacity DM must tackle large scale, high-dimensional data such as those encountered in video generation. Lastly, TADA currently uses 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.
10

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[18] Tero Karras, Miika Aittala, Jaakko Lehtinen, Janne Hellsten, Timo Aila, and Samuli Laine. Analyzing and improving the training dynamics of diffusion models. Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 24174-24184, 2024.
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[19] Diederik Kingma and Ruiqi Gao. Understanding diffusion objectives as the elbo with simple data augmentation. Advances in Neural Information Processing Systems, 36:65484-6553 2023.
[20] Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747 2022.
[21] Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint (rXiv:2209.0303: 2022.
[22] Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775- 5787, 2022.
[23] Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm- solver++: Fast solver for guided sampling of diffusion probabilistic models. arXiv preprint arXiv:2211.01095 2022.
[24] Dustin Podell, Zion English, Kyle Lacey, Andreas Blattmann, Tim Dockhor, Jonas Müller, Joe Penna, and Robin Rombach. Sdxl: Improving latent diffusion models for high-resolution image synthesis. arXiv preprint arXiv:2307.01933 2023.
[25] Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint
2022.
[26] Simo Särkkä and Arno Solin. Applied stochastic differential equations, volume 10. Cambridge University Press, 2019.
[27] 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. 134466, 2020.
[28] Yang Song, Prafulla Dhariwal, Mark Chen, and Iya Sutskever. Consistency models. In International Conference Machine Learning, pages 02211-32252 PMLR, 2023.
[29] George Stein, Jesse Cresswell, Rasa Hosseinzadeh, Yi Sui, Brendan Ross, Valentin Villecroze, Zhaoyan Liu, Anthony 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, 36:3732-377 2023.
[30] Daniel Watson, William Chan, Jonathan Ho. and Mohammad Norouzi. Learning fast samplers for diffiusion models by differentiating through sample quality. In International Conference on Learning Representations, 2021.
[31] 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:2306.0933 2023.
[32] Shuchen Xue, Mingyang Yi, Weijian Luo, Shifeng Zhang, Jiacheng Sun, Zhenguo Li, and Zhi- Ming Ma. Sa-solver: Stochastic adams solver for fast sampling of diffusion models. Advances in Neural Information Processing Systems, 36:77632-77674, 2023.
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12

---

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, := A @ la.
p4 Ma Id Lie Liv & La 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.
min (0) = -,ex,,celllo0xx 10000-(())) e01
-x
[ eo(xxxx) - - 100
(7)
T
1 (0)
1 T
= -En et L
1 co(xx,)) - 1t
(1)
It
14 (0)
(1)
I1
11
L
L
(8)
L
L
= =zznei Lin?
| (x((xtt) t) - -x +
L Mt
(1)
ke (0)
T1 11
(9)
L
L
L
14
(1)
1ee (0)
Lo (xx,
+
L
It
(0)
-x,
12
(10)
L
Les
14
parameterized Neural Netowrk
Following the same spirit, one can derive the case for variable. See Appendix. | for details. We know that,
X I1 ~ N( Ex),
Define
Σx
1/ = -r],.
11 TE, ue
and the residual ("noise""
c = X4
Vt ft.
Since the operation is linear, one can transform the X₃ by
T: (Nt, E) = (r, X1, X x.)
and it is invertible with linear inverse 3ef4 Hence the o-algebras coincide:
a axx,) (v..)) 6).
For any integrable random variable Z, equal a-fields imply E|Z xx] = E[Z yr, E). Taking Z = yields
E xx] E[ y/s, e).
due to the fact that is the independent gaussian, thus
Biz
xx]
E[x1
Ve, c] E[ ye, = E[zr vx]
13

---

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
dx;
A₁ Xt + Fr,
To ~ ~0,, 7)-
(11)
dt
and again, in this paper, only consider,
0
1
0
0
0
1
0
A,
:
and
b,
(12)
0
0
0
0
0
NxN
If expand the system, it basically represent:
dx. (0) x;
(1) dt
(13)
dx, (1) z! dt
(14)
(15)
dx (N-1) 1) F,dt
(16)
(17)
In our case, the F; function is:
F:= N! 21
N-1 st Zi=
2. (1
- t)
(18)
1)
One can easily verify that, when = 1, it actually degenerate to flow matching:
T1 Xx
F.
and
(19)
1 1
dæ. = F,d
(20)
One can consider it as the higher augment dimension extension of flow matching model. And the magical part that, do not need retrain model. For better analysis, we rearange the system:
dx, dt
A.xi b.Fe
(21)
N-1 (4)
= A, + N.N! 31
1=1 21 (1 - t)²
(22)
1).
N-1 (9)
N
b
D.NIEN E
(1 t)
A +
(23)
(1 - t)
- t)
0
0
0
= A
F1
(24)
N!
N1-1)
(1-1)*NI
NI
(1-1)
01(1 01(1-1)
11(11)18
(N (N-1)
b.
A,x, + b
(Å, = A, - -A)
(25)
14

---

Consider the linear time-varying system:
dxt dt
A. b. x1, Xo ~ (0)
(26)
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
m, E[x,].
Then
m
A, m, + be 21.
mo = 0.
(27)
We can write the mean in factorized form as
mr p0z
so that by dividing by Tı, we obtain
14 Arpe + bt.
Covariance Dynamics. Similarly, the convariance follows dynamics
E, A? Σ = A E, + E,
(28)
Recall that, Then we define the scalar quantity of interest as
T
T
Er mm
Xt
Σx 14
Xt
r X
=
with Te = u4 E, p4
(29)
-1 me Σ, Prt.
fe. D,
7t
We wish compute the derivative of Using the quotient rule.
r!'xt
yt
7t
d
(r. X1)
9t
(r.
4
2
Since r!'x, x; = Yes this becomes
1 X
+
2 dt
Yt 7tt
31
7t.
716
Derivative of r,
Recall that
= E,
Diffcrentiating gives
f -, 15, Σe fb Es -1 -
Substitute the known dynamies:
Σ A. + E, AT.
14: d. + b
It follows that
f; = -;
- Ej E, -1 pr + E; -1 jir
(30)
=-E,- A E₁ + Σ, A?
Σi Mc + E. -1 (Ai +bi)
(31)
--E, -E A.T. nu E, "ippe + E, -.
(32)
=-ATr; + E, -,- -b...
(33)
15

---

Derivative of X,
From 26).
X Aix: + 21.
Derivative of 7/.
Recall
7t pel D1 pe re Pe.
Differentiating.
8 & r pr.
Using
30 and pi, Aep:: + bi, one obtains (after cancellation) the result:
(-ATr, + E,
-,,,,,) 'b,) Me +r (Arpee + b.)
(34)
2 rt
(35)
2 2 Ee 14
(36)
Combining Everything
Substitute the pieces into
X + r
Xt
20 7
Vt
7b.
Using
* = rf A, + bjf E,
r x; r² (Axx, + +b.
we have:
1 x4 + r/'x
[ A 63 Σ,
x + [ + rx Be
=b; En x; r b
Thus,
X, + I1 r b
yt xt
Ú.
(37)
Vt.
Recall that 7/6 me T E," Me and i/1 2b? E, -1
Me. Also note that
rf b. ue Σ, b
Thus, the final expression becomes
b/ E, -1
X2 1 Me T TS -1 b
2b. E, -1
Hi
1/
U/t
(38)
Me I' Σx
M1 T' E, p4
6/'' E, -1
M1 T E, -1 by
2 63 E; ktt
+
(39)
14 Σ,
11 T
14
11, Σ, f4
The frist term is essentially one kind of linear combination of x,, and Recally that y/ = rrxx x, := -. which is another lincar combination of x;. Assume that x, ~N(pa Ee), thus, one can E. n. derive the relationship between the frist term and ys. Thus, according Lemma. I.
by7T Σ, 4 Σ, -1 b1
Era. E.r,
e. e.Err;
Xi e I
Mad +
y1 + e LEEL,
(40)
r. ETE
T1 ~ N(O, la L, L. L...L./r.. Err)
(41)
16

---

Thus, by plugging in the expression, one can get:
j/ = Cztlt + Brn + e T
(42)
2 auye + Bro + LieT
(43)
Where,
0 Err
2b E. -1 at
(44)
r Erre
H4 Σ," ft
8t Pi T Eg
b + e
Ex1r
I
Mt
(45)
11 T Σ,
r Egr
W
(4)
(e, (1) (7)
(46)
D Experiment Details
Here we elaborate more on experiment details.
D.1 EDM and EDM2
For the baselines on EDM and EDM2 codebase, we directly use the code provide in DPM-Solver-
v3 [36]. For the fair comparision, for all baselines, controlled min 0.002 and max 80 as suggested in the original EDM and EDM2 paper.
For DPM-Solver++ we did abalation search over order E [logSNR, time uniform, edm, time quadratic].
[1, discretization
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 V2. to evaluate generate dresults.
E
Additional Plots
E.1
General N variable dynamics
This section referenced in the main paper and will be removed soon; is retained only for now to keep the appendix numbering aligned with the main paper.
F Detailed Explanations
F.1 Explicit form of A, and
Here we demonstrate the A, and b, 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 derived which is straight-forward. We know x (0) the position and define higher derivatives recursively
: I, be
dlrr(0) (0)
(K)
k 1, I.....N
dtk
17

---

Table Comparison of different solvers
Algorithm
A
b
Fj
AGM
[0, 1]"
21-x (0)
1-1 X
(1)
0
-M11
M
CLD
B
[0,1
( log p(2,
TM-1
The system dynamics form an Nth-order chain of integrators driven by scalar input F(t,x,) x,):
1 2;
(1)
(1) (2)
1t Ft
2 (N (N-1)
21
(N-1) F(t.x))
(47)
Equivalently, the position satisfies the scalar ODE
d (0)
dt Ft,,
Our goal starting at some time (0,1) 1) with known state (x(²)) k== position reaches prescribed value at / = 1:
(K) N-1
choose F so that the
("hit the target").
Assume is held constant the remaining interval [t, 1]. Repeated integration yields the degree-N Taylor polynomial about l:
N-1
(1 - t)N N F.
(R)
21
(48)
k!
k=0
Thus, one can simply solve the F by Rearranging
48
to isolate F:
N!
N-1
(R) k=0
[1 t)*
F
k!
Thanks to the simple form of F, one readily write down the mean and covariance of the system. Now we know
0
1
0
07
0
0
1
0
A =
b
10....,0,11] 1
0
[0
0
By rearraging the dynamics, gives the linear time-varying closed loop
X
(49)
w/ control
b w/ control
We need first compute the transition matrix induced by Å, and we call it controlled transition matrix. By Solving + 4 column-wise gives the polynomial matrix
18

---

t m
NItN-4
2 k.
(t. 0) (Tem (t) N-1
km (t))
Tk (t)
(m k)!! N-k (N km!
(N k)lm!
m < k.
Plugging this (tt, 0) into the boxed formulas above supplies p(t) and (1) explicitly for every order
N.
Let ((t) B[x,]. Because 1 is deterministic,
A(() N(1) + b, 21,
u(0) = po
(50)
Define the state-transition matrix (), of Aci
(b,, A, (b,, T), P(T,T) I.
Then the standard variation-offcconstmm formula gives
[." (tt, r) b, xı dr.
F = [tt,, 0) po +
If the initial derivatives are i..d. AC 1). then pr 0 and only the integral term remains. Carrying out the integral (polynomials of r) yields
(k) N! H4 (N k) 21,
k= O,...,, -
Meanwhile, propagation of covariance matrix is:
Σ A,E, + E, 2()) Eg.
(51)
Eq. 51 is homogeneous Lyapunov ODE whose unique solution is exactly (see Appendix, more details):
H for
Σ, (tt, Eg (t. T)
F33
Previous Fast Solver
Table 2: Comparison different solvers
Order type
Order
Multistep type
Expansion term
Discretize space
Heun
Single Step
2
N/A
N/A
DEIS
33
Multi-step
2/3/4
E8
DPM-Solver 22
Multi/Single-step
2/3/4 /
Adams-Bashforth
6Ø
Optional Optional Optional
DPM-Solver++
23
Multi/Single-step
2/3/4
Adams-Bashforle
Z8
UniPC 34
Multi-step
3/4/5
Adams-Moulton
TouS(ours)
Multi-step
2/3
Adams Bashforth
Fo
t
F44
Extended Flow Matching
In the framework of flow matheing, obtain the velocity by T-r because it is the linear interpolation between data and prior øn- And meanwhile, it happens to be the solution of optimal control problem:
min S.
||v. [1x s.t dx, w,tt
(52)
19

---

For the detailed derivation, please see Sec.C.1 in [6] For AGM, they consider momentum system, which reads
min J. [oe|Bae, s.t dæ, v,dt, duz - a,dt + dwy
(53)
The differences is that, AGM consider the injection of stochasticity in the velocity channel. For our case, the Fa derived in Sec. 3.2 is the solution for
min / || F. d2 Fi
dx, (0) xi ( dt
drt xt dr
dr; (N-1) F,d
and its spirit keeps as previous formulation, move Si to x Pdata from ( t / 1.
(0)
F,5 Degenerate Case of TADA
Here we discuss about the degenerated case of TADA. The reasoning behind it rather simple. We show the dynamics of (eq. 39) again here,
67
Σ
H1 T Σ, b,
2b E;
Mt pb.
11
pr
X
He T D,
-1 pt
T1
na E,
(54)
and recall that
14 Σ
Vt
(55)
p4 T
Thus, if the first term depends exclusively on the system reduces to the scalar ODE for y and becomes formally identical to other diffusion-model parameterizations such VP, VE, FM. More precisely, in order to degenerate TADA, one only requires
[ b;
(56)
where b, is defined in eq. 24 This proportionality holds in two scenarios:
1. When N = 1. so that and b, are scalars. In that case, the framework collapses to flow matching- mere reparameterization the diffusion model.
When A, is diagonal and its components evolve independently. Then every dimension of μ4 and b, shares the same mean and variance, and proportionality follows directly.
A simple empirical check to propagate the model from different random initializations using formulation: yields identical FID scores after generation, confirming the degeneracy.
G Additional Qualitative Comparision
Please see fig.
10 and fig.
H Solution of the homogeneous Lyapunov ODE
Let (tt,7) be the state-transition matrix of the (possibly varying coefficient Ag:
(t. r) A, T),,T),
Φ(T,T) T.
Throughout we abbreviate (tt, 0) = (tt).
20

---

MSD(ours)
UniPC
MSD(ours)
UniPC
UniPC MSD(ours)
Figure 9: Additional Visual comparision with UniPC using EDM2 w/ 5 NFEs.
Candidate solution.
Consider
Σ(I) Φ(1) Eo D()
2. Verification. Differentiate previous equation and use the product rule together with éb() AI)(() and de (() +b(() A7:
(t) = EG + 1 4
= + A7
AtE(() + Et)
Hence E(t) satisfies the differential equation in (Lyap), and Σ(0) (()) Eg.
3. Uniqueness. Lyapunov Equation is linear in the matrix variable D; by the Picard-Lindelöf theorem its solution is unique. Therefore (S) the solution.
I Correlation between two Gaussian Variable
Lemma I.1. Let the random vector
Xa ~ Mp 21, E.):
Σ, = L,L.
(Cholesky factorisation).
For two fixed column-vectors r, E Rd set
V1 : X
:=== e.
21

---

MSD(ours) )
UniPC
MSD(ours)
UniPC
MSD(ours)
UniPC
Figure 10: Additional Visual comparision with UniPC using EDM2 w/ 5 NFEs.
Write the convenient abbreviations
C: = r, p4 31,
d= 21,
g: L,r., h L.er, of o -l
Then
Egrrr T r E,r:
of Err; r E,r
21
T
I
p431
Yt L.LL,
< Ia L. E.r.).
Proof.
+ g
== de h,,
Any vector can be decomposed into the component along and the component orthogonal
g
(g.
EL.
g g|e =0
Because e (0, Ia) and the projector onto g: is orthogonal to the projector onto the complement, g. 6 and 6 are independent Gaussian variables.
Insert g, E y/ C obtain
g:
(y: c)) EL.
El ~M,000 la 1gg.g/////
y
22

---

Then we can plug this into 2y:
2x d, + h!
gr
(yt c) + EL)
(57)
o
h g:
e p1i
(1/t
ct) + +h,e
(58)
h gt /1
h h. ue h
c. pm1
(59)
e, Err
e E
e, 1 +
(r] Iı + LEET
(60)
r Exr,
r
Egrer
0. Eert
e,
T
I
M4 +
y1 + LieT
(61)
r Err
r. Err
J General N variable MDM loss
At time the N-variables are generated by
X H4 21 L
E~ N~N(0, IN):
with known p4 RN and invertible L, E NNN The goal is to predict c(N-1))
By whitening trick, one can isolate (N-11), Let e (O...., 0, be the row vector that selects the last coordinate. Left-multiplying by e gives
eN L, X1 c L,, 1 + CN-1 L, L,e.
IN
Define the time-dependent scalars
a == eN RN,, b-:= a aT pe +
then
(N-1)) = a, by x1-
(62)
A neural network co(x;, f) is trained approximate c(N-1) with the standard
LMDM (0) == E (x(x,, (2
Insert eq. 62 and multiply the interior by by:
(MD (X t) 1 a. xx b x
x (xe, t) 22113
where
go(x1, 6x 1) a. Xt.
be
23

---

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Justification:
We have a limitation section in the end.
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Reviews:
[{"review_id": "inALwtiKL0", "reviewer": "Reviewer_P5Q9", "summary": "- 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", "questions": "- 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.", "limitations": ".", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## 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).", "was_revised": true, "final_justification": "To sum up,\n\n- The proposed approach is novel\n- The experimental results are promising\n- The authors have addressed my concerns during the rebuttal phase\n- Inconsistencies I previously identified were acknowledged and the authors have committed to implementing the necessary corrections in the final version of the paper", "rebuttal": "# Rebuttal\n\nWe thank the reviewer for thorough comments on both the main paper and the appendix. Below are our clarifications.\n\n---\n\n### 1. Lines 30–33: Statement is overly strong  \nAgree. This should be the conditional statement.We will revise it to:\n\n> “... 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.**”\n\n---\n\n### 2. Shape mismatches in Appendix A  \nThe shapes are correct because we abused the notation (sorry!)  for all coefficient functions of $t$ by implicitly expanding them with the Kronecker product for simplicity. This is stated at the first line of the appendix. However we now recognize this is easy to miss and can lead to confusion. To remedy this, we will superscript all such variables with $\\otimes$, e.g. $\\mu_t^\\otimes=\\mu_t\\otimes I_d$.\nDoes this address your concerns?\n\n---\n\n### 3. Redundancy of the two state equations  \nThis redundancy questions the need for an auxiliary variable during **training**. Our experiments, which sample from the momentum system using the pretrained diffusion model, confirm this observation, which means the auxiliary variable is indeed redundant during **training**.\n\nSince the auxiliary variable truly offers no benefit during training, the next question is whether the momentum system improves sampling. Should the two formulations (momentum sampling vs vanilla diffusion sampling) remain mathematically identical, there would indeed be no justification,per the reviewer’s observation,for introducing an auxiliary variable. \n\nYet, as Proposition 3.3 shows, the momentum system solves an ODE with SDE-like diversity, which can potentially enhance sampling quality (i.e see Fig. 5b–c in the EDM paper and Table 1 in the DDIM paper). And we further verify it in Proposition 3.3 and the experiment section.\n\nTo summarize, we agree with the reviewer that introducing an auxiliary variable yields no benefit during training, as already advocated in the paper. However, this makes a meaningful difference during sampling.\n\nWe will emphasize this again in the introduction for clarity.\n\n---\n\n### 4. Exact expression for $\\mathbf{r}_t$, $\\mathbf{\\mu}_t$ and $\\mathbf{\\Sigma}_t$   \nAll closed-form expressions are provided in Appendix E.2. Here, we walk through them to aid the reviewer’s understanding and ensure clarity: \n\n1. Given number of variable $N$, user can simply compute the $\\mu^{(i)}_t$. \n\n2. $\\mathbf{\\Sigma}_t$ is more complicated. We derive it from $\\Phi(t,0)\\mathbf{\\Sigma}_0 \\Phi(t,0)^T$ and $\\mathbf{\\Sigma}_0$ is the prior covariance matrix which is user's choice. The explicit expression of $\\Phi$ is provided in Appendix E.2.\n\n3. Here is a quick example for $N=2$ case. By simply plugging $N$ and $t$ in equation, we get $\\mu_0^{(0)}=t^2x_1$,$\\mu_t^{(1)}=2tx_1$, $\\Phi(t,0)=[[1-t^2, t-t^2],[-2t,1-2t]]$. Asumme $\\Sigma_0 = [[1,0],[0,1]]$, Then $\\Sigma_t=\\Phi(t,0)\\Phi(t,0)^{T}$. Then the user are ready to compute SNR and $\\mathbf{r}_t$ by using equations above.\n\n4. After obtaining $\\mathbf{\\mu}$ and $\\mathbf{\\Sigma}_t$, the user can get $\\mathbf{r}_t$ by $\\mathbf{r}_t=\\frac{\\mathbf{\\Sigma}_t^{-1}\\mathbf{\\mu}_t}{\\mathbf{\\mu}_t^T\\mathbf{\\Sigma}_t^{-1}\\mathbf{\\mu}_t}$, meanwhile the effective SNR is the denominator $\\mathbf{\\mu}_t^T\\mathbf{\\Sigma}_t^{-1}\\mathbf{\\mu}_t$, and the user can use this value to match the SNR of Flow matching and Diffusion Model.\n\n---\n\n### 5. Takeaway from Fig.2   \nHere are the motivations of Fig.2:\n\n1. We want to emphasize that our model can produce diverse outputs from a single initial state, exhibiting SDE-like diversity—albeit without true stochasticity. Specifically, the diversity comes from the interaction of $N$ variables which is theoratically proved in Prop.3.3 and we name it pseudo-noise.\n\n2. We include Fig.2 to demonstrate that our method is not merely an alternative time-discretization of the vanilla diffusion model. If it were, the generation trajectories along the SNR (or effective-noise) from the same starting point—would coincide with EDM-ODE. The fact that they do not overlap underscores the unique dynamics of TADA, which in turn contributes to improved generation performance.\n\n---\n\n### 6. Arbitrary choice of NFE for Heun?   \n511 NFE is the default value used in EDM paper for Stochastic sampling and we notice that this number performance best. Applying the same logic to EDM2, we use the 63-NFE schedule cited as that paper’s default;\n\n---\n\n### 7 Summary\nWe hope this rebuttal can more or less address your concerns. Please feel free to raise any additional questions! We view the rebuttal process as a valuable opportunity to improve the quality of the paper and to validate that our ideas and results are meaningful to the broader community.\n\nAlso, If all concerns are resolved, we would deeply appreciate a reconsideration of the overall rating :)"}, {"review_id": "h3hWaBXnN2", "reviewer": "Reviewer_W3UK", "summary": "This 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.", "questions": "I 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.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## [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.", "was_revised": true, "final_justification": "I believe this paper will be beneficial to the diffusion model community, primarily due to the strengths I previously mentioned. Specifically, the authors demonstrate the equivalence between the momentum diffusion model and conventional diffusion models through sound theoretical analysis. This shows that the advantages of the momentum diffusion model, despite its training instability, can be leveraged in a training-free manner using models trained with the traditional diffusion loss. This approach is both reasonable and novel, especially considering that similar ideas have been observed in prior diffusion samplers.\n\nHowever, as noted by myself and other reviewers, the paper contains many unclear sections that require considerable effort to fully understand. Improving the clarity of the writing is essential. If other reviewers or the Area Chairs feel that these clarity issues cannot be resolved through revision, I am willing to defer to their judgment. Otherwise, I recommend that the authors significantly revise the manuscript to improve its readability and accessibility.", "rebuttal": "# Rebuttal\n\nWe thank the reviewer for thorough comments on both the main paper and the appendix. Below are our clarifications.\n\n---\n\n### 1. CLIP and FID on COCO  \nWe still have marginal improvement in terms of FID on COCO validation set especially for small NFE. We found that standard CLIP tends to favor blurry images as long as they are semantically aligned with the text, which can result in inaccurate evaluations. Instead, we chose to report HpsV2, a fine-tuned variant of CLIP, as it offers more reliable and meaningful assessments in our setting. if the reviewer believes that including the standard CLIP score is necessary, we are happy to incorporate those results in future discussions.\n\n\n|  FID-COCO-CFG1/NFE         | 6     | 10      |15|\n|-----------|-----------|-----------|-----------|\n|UniPC| 25.15 |19.56   |20.61|\n|TADA| 21.64| 19.85 |  20.46|\n\n|  FID-COCO-CFG2.5/NFE         | 6     | 10      |15|\n|-----------|-----------|-----------|-----------|\n|UniPC| 16.97|15.20  |  14.92|\n|TADA| 16.68| 15.20 |  14.91|\n\n|  FID-COCO-CFG3.5/NFE         | 6     | 10      |15|\n|-----------|-----------|-----------|-----------|\n|UniPC| 17.44|15.78  |  15.47|\n|TADA| 17.35| 15.78 |  15.46|\n\n\n---\n\n### 2. Lack of Definitions in Fig.1  \nThanks for the feedback. We will add explanations for each variable in the caption and also hyperlink to the exact equation to increase the accessibility.\n\n---\n\n### 3. Align the formulations of Eq. (3) and the MDM training objective in the proposition.  \nWe are going to switch all the notation from $\\epsilon$ prediction to $x_1$ prediction.\n\n---\n\n### 4. Indicate force reconstruction  \nThis is indeed a good suggestion for clarity of the paper! We will have explanation on how to reconstruct $F_{\\theta}$ in Prop.3.1 and also provide a reference to section 3.2.\n\n---\n\n### 5. Lack of Explanation of Prop.3.3  \nGood catch! We sincerely appreciate the reviewer’s effort in working through the detailed Appendix B. Our goal was to keep the main paper accessible by moving most theoretical material to the appendix, but we now see that the main text may be overly concise. \n\nWe will therefore, at least, add an intuitive explanation of $\\epsilon^{(i)}$ to the main paper.\n\n---\n\n### 7. Inconsistency references format. \n\nWe sincerely thank the reviewer for pointing out this seemingly minor but significantly important point for readability. We will make sure to address it in the revision!\n\n---\n\n### 8. Welcome to join discussion\nWe further provide more interesting experiment results and analysis during the discussion with other reviewers. Feel free to have a look if you are interested in!"}, {"review_id": "DNuqiQfVr4", "reviewer": "Reviewer_w9nZ", "summary": "This 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.", "questions": "1.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?", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "While the authors' rebuttal provided helpful clarifications, core concerns persist:\n- Clarity is insufficient\n- The paper lacks essential ablation studies\n- The authors provide no convincing explanation for the underlying mechanism \n- CIFAR-10 failure contradicts the claimed \"SDE-like properties\" advantage   \n\nAcknowledging the authors' efforts, I adjust my rating from 2 to 3.", "rebuttal": "# Rebuttal \n\nWe thank the reviewer for thorough comments on both the main paper and the appendix. Below are our clarifications.\n\n---\n\n### 1. Computation Complexity and memory complexity  \nboth computation and memory complexity overhead are marginal.\n#### Computation complexity\n1. All computations depend solely on time and can be precomputed after discretizing the time horizon, prior to sampling.\n2. Although operations involving $\\mathbf{\\mu}$ and $\\mathbf{\\Sigma}$ may look costly due to the Kronecker product with identity, the effective dimension is $N$ (the number of variables). Thus, we only need to store and compute with $N$-dimensional matrices/vectors.\n3. The major computation, which is the Neural network, has the same input and output dimension, which is identical as baselines.\n\nOverall, the computational overhead arises primarily in points 1 and 2, and both are theoretically and empirically negligible. We further validate this experimentally by measuing throughput speed on ImageNet with 10 NFE. As the network size increases the performance gap further narrows:\n|           | UniPC     | TADA      |\n|-----------|-----------|-----------|\n| ImageNet64| 0.0029 sec/Img  | 0.00324 sec/Img |\n| ImageNet512| 0.0087 sec/Img  | 0.0090 sec/Img |\n\n#### memory complexity\nAnd here is practical memory usage for Imagenet 512 w/ Batch=64:\n|  N         | 1     | 2      |4|8|\n|-----------|-----------|-----------|-----------|-----------|\n|Memory-GB| 16.96|16.99  |  17.01|17.11|\n\n---\n### 2. Missing ablation study for TADA  \nHere are ablation experiments, (we skipped order=3 since it has been reported in the paper)\n|      ImageNet512/NFE     | 10     | 13      |15|21|\n|-----------|-----------|-----------|-----------|-----------|\n| UniPC-order-1| 17.11  | 8.01|5.62|3.10|\n| TADA-order-1| 12.51  | 7.08|5.45|3.53|\n| UniPC-order-2| 4.80  | 2.97|2.54|2.14|\n| TADA-order-2| 2.83  | 2.35|2.21|2.09|\n| UniPC-order-4| 8.68  | 2.86|2.12|1.99|\n| TADA-order-4| 3.83  | 2.10|2.00|1.97|\n\n---\n### 3. Better Quality like SciRE-Solver   \nYes. see right side of Fig.5.\n\n---\n\n### 4. Mechanistic understanding? Cifar10 experiments?  \nWe address both questions jointly because the CIFAR-10 experiments partially verify the proposed mechanism.\n\nOn small-scale data such as CIFAR-10, once a high-capacity model is well trained (e.g., EDM), ODE sampling performs better than SDE sampling, regardless of the amount of injected noise (see Fig. 5b in the EDM paper). By contrast, in large-scale settings (more data and higher dimensionality) such as ImageNet,where training an equally strong model is harder, SDE sampling tends to outperform ODE sampling (see Fig. 5b–c in the EDM paper and Table 1 in the DDIM paper) when NFE is sufficient.\n\nOur hypothesis is that TADA does not provide a fast-sampling mechanism. Instead, it leverages stochasticity to compensate for propagation inaccuracies just like SDE, though it is pseudo-noise/stochaticity. Since SDEs are often slow due to their sensitivity to time discretization, the pseudo-noise in TADA offers a practical middle ground: retaining SDE-like behavior while enabling low NFE sampling through an **ODE** solver, thanks to pseudo-noise.\n\nThis aligns with our experiments: TADA consistently outperforms baselines on large-scale ImageNet and SD3, and as expected, underperforms ODE sampling on CIFAR-10 with the EDM checkpoint. The results below illustrate this pattern.\n\n|      NFE     | 10     | 13      |15|21|\n|-----------|-----------|-----------|-----------|-----------|\n| TADA| 3.4  | 2.58|2.32|2.04|\n| UniPC| 2.4| 2.05 |1.96|1.91|\n\nThis further supports our recommendation in Sec.5 to apply TADA to larger models, rather than focusing on small-scale ones.\n\nWe can highlight this property in the paper to provide more intuition!\n\n---\n\n\n### 5. 186% Faster claim  \nIt is computed as the number of NFE used to achieve FID=2 in imagenet512. \n\nTADA converges to FID=2 at NFE=12. \n\nAs Figure 6 shows, the other solvers never dip below FID 2 within 35 NFEs. To dig deeper, we ran a full ablation and found one UniPC setup (row 7 in the EDM2_unipc_ablation file) hits FID 2 at roughly 23 NFEs, about a 186 percent improvement. \n\nHowever, This configuration falls well behind the paper’s version when the NFE budget is small (NFE in the range 5 to 20, see rows 2–6 in EDM2_unipc_abaltion file).\n\nIn short, we searched every UniPC variant and picked the one that reaches FID 2.0 with the fewest NFEs. We hope that is the fairest way to compare.\n\n\n### 6. Why change title?   \nWe found this title sounds better :)\n\n### 7 Summary\nWe hope this rebuttal can more or less address your concerns. Please feel free to raise any additional questions! We view the rebuttal process as a valuable opportunity to improve the quality of the paper and to validate that our ideas and results are meaningful to the broader community.\n\nAlso, If all concerns are resolved, we would deeply appreciate a reconsideration of the overall rating :)"}, {"review_id": "ycRdDZCekO", "reviewer": "Reviewer_vgnG", "summary": "This 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).", "questions": "see weakness", "limitations": "yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "I believe this article has a good approach and will bring some new perspectives to the community. However, I hope the author can more clearly revise some of the expressions in the article.", "rebuttal": "# **Rebuttal*\n\n### 1. larger value for N in SD3?   \nGreat question! We did conduct this experiment with SD3 and observed that using N = 4 resulted in even faster convergence. However, unlike the N = 2 case, we were unable to find a configuration that consistently outperformed the baseline in EDM2, as N = 2 did. To maintain consistency in presentation, we chose to omit the N = 4 results. Our objective is to provide a robust sampling method that performs reliably without requiring extensive tuning. Since you brought it up, here is the list of results for SD3 with N = 4 and it performs better for CFG > 1 case:\n|     CFG=1/ NFE     | 10     | 20      |30|\n|-----------|-----------|-----------|-----------|\n| N=2| 24.56 | 24.85|25.01|\n|N=4| 23.64| 24.23 |24.38|\n\n|      CFG=2.5/NFE     | 10     | 20      |30|\n|-----------|-----------|-----------|-----------|\n| N=2| 29.73 | 30.10|30.18|\n|N=4| 30.03| 30.17 |30.22|\n\n|      CFG=3.5/NFE     | 10     | 20      |30|\n|-----------|-----------|-----------|-----------|\n| N=2| 30.2 | 30.72|30.81|\n|N=4| 30.47| 30.83 |30.90|\n\nWe are happy to present this in the appendix as well.\n\n### 2. Lossless approximation for coefficients   \nYes. Everything is exact and has a closed-form solution. The only approximation is for $x_{\\theta}$ (data) prediction which is using a Neural Network as is classical for diffusion.\n\n### 3. Estimation error for F in larger scale dataset   \nThere will be no approximation error for $F_{\\theta}$. See above clarification.\n\n### 4. Welcome to join discussion  \nWe further provide more interesting experiment results and analysis during the discussion with other reviewers. Feel free to have a look if you are interested in!"}]

Author rebuttal:

Meta-review: This work proposes TADA (Training-free Augmented DynAmics), which can take pre-trained diffusion models such as stable diffusion and adds higher dimensional momentum terms during inference time. This reduces the number of function evaluations required to generate high-quality samples on models such as EDM and Stable Diffusion. The reviewers appreciated the detailed experiments and the mathematical derivations. The concerns raised by the reviewers were also satisfactorily answered by the authors during the rebuttal. I recommend acceptance.
Decision: Accept (poster)