# constant_acceleration_flow__b43f2b39.pdf

Constant Acceleration Flow

Dogyun Park Korea University gg933@korea.ac.kr

Sojin Lee Korea University sojin_lee@korea.ac.kr

Sihyeon Kim Korea University sh_bs15@korea.ac.kr

Taehoon Lee Korea University 98hoon@korea.ac.kr

Youngjoon Hong KAIST hongyj@kaist.ac.kr

Hyunwoo J. Kim Korea University hyunwoojkim@korea.ac.kr

Rectified flow and reflow procedures have significantly advanced fast generation by progressively straightening ordinary differential equation (ODE) flows. They operate under the assumption that image and noise pairs, known as couplings, can be approximated by straight trajectories with constant velocity. However, we observe that modeling with constant velocity and using reflow procedures have limitations in accurately learning straight trajectories between pairs, resulting in suboptimal performance in few-step generation. To address these limitations, we introduce Constant Acceleration Flow (CAF), a novel framework based on a simple constant acceleration equation. CAF introduces acceleration as an additional learnable variable, allowing for more expressive and accurate estimation of the ODE flow. Moreover, we propose two techniques to further improve estimation accuracy: initial velocity conditioning for the acceleration model and a reflow process for the initial velocity. Our comprehensive studies on toy datasets, CIFAR-10, and Image Net 64 64 demonstrate that CAF outperforms state-of-the-art baselines for one-step generation. We also show that CAF dramatically improves few-step coupling preservation and inversion over Rectified flow. Code is available at https://github.com/mlvlab/CAF.

1 Introduction

Diffusion models [1, 2] learn the probability flow between a target data distribution and a simple Gaussian distribution through an iterative process. Starting from Gaussian noise, they gradually denoise to approximate the target distribution via a series of learned local transformations. Due to their superior generative capabilities compared to other models such as GANs and VAEs, diffusion models have become the go-to choice for high-quality image generation. However, their multi-step generation process entails slow generation and imposes a significant computational burden. To address this issue, two main approaches have been proposed: distillation models [3, 4, 5, 6, 7, 8, 9] and methods that simplify the flow trajectories [10, 11, 12, 13, 14] to achieve fewer-step generation. An example of the latter is rectified flow [10, 11, 13], which focuses on straightening ordinary differential equation (ODE) trajectories. Through repeated applications of the rectification process, called reflow, the trajectories become progressively straighter by addressing the flow crossing problem. Straighter flows reduce discretization errors, enabling fewer steps in the numerical solution and, thus, faster generation.

Rectified flow [10, 13] defines the straight ODE flow over time t with a drift force v, where each sample xt transforms from x0 π0 to x1 π1 under a constant velocity v = x1 x0. It

Corresponding authors.

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Flow crossing

Sampling trajectory

(a) Rectified Flow

Flow crossing

Sampling trajectory

(b) Constant Acceleration Flow

Figure 1: Initial Velocity Conditioning (IVC). We illustrate the importance of IVC to address the flow crossing problem, which hinders the learning of straight ODE trajectories during training. In Fig. 1a, Rectified flow suffers from approximation errors at the overlapping point xt (where x1 t = x2 t), resulting in curved sampling trajectories due to flow crossing. Conversely, Fig. 1b demonstrates that CAF, utilizing IVC, successfully estimates ground-truth trajectories by minimizing the ambiguity at xt.

approximates the underlying velocity v with a neural network vθ. Then, it iteratively applies the reflow process to avoid flow crossing by rewiring the flow and building deterministic data coupling. However, constant velocity modeling may limit the expressiveness needed for approximating complex couplings between π0 and π1. This results in sampling trajectories that fail to converge optimally to the target distribution. Moreover, the interpolation paths after the reflow may still intersect a phenomenon known as flow crossing which leads to curved rectified flows because the model estimates different targets for the same input. As illustrated in Fig. 1a, instead of following the intended path from x1 0 to x1 1, a sampling trajectory from Rectified flow erroneously diverts towards x2 1 due to the flow crossing. Such flow crossing can make the accurate learning of straight ODE trajectories more challenging.

In this paper, we introduce the Constant Acceleration Flow (CAF), a novel ODE framework based on a constant acceleration equation, as outlined in (4). Our CAF generalizes Rectified flow by introducing acceleration as an additional learnable variable. This constant acceleration modeling offers the ability to control flow characteristics by manipulating the acceleration magnitude and enables a direct closed-form solution of the ODE, supporting precise and efficient sampling in just a few steps. Additionally, we propose two strategies to address the flow crossing problem. The first one is initial velocity conditioning (IVC) for the acceleration model, and the second one is to employ reflow to enhance the learning of initial velocity. Fig. 1b presents that CAF, with the proposed strategies, can accurately predict the ground-truth path from x1 0 to x1 1, even when flow crossing occurs. Through extensive experiments, from toy datasets to real-world image generation on CIFAR-10 [15] and Image Net 64 64, we demonstrate that our CAF exhibits superior performance over Rectified flow and state-of-the-art baselines. Notably, CAF achieves superior Fréchet Inception Distance (FID) scores on CIFAR-10 and Image Net 64 64 in conditional settings, recording FIDs of 1.39 and 1.69, respectively, thereby surpassing recent strong methods. Moreover, we show that CAF provides more accurate flow estimation than Rectified flow by assessing the straightness and coupling preservation of the learned ODE flow. CAF is also capable of few-step inversion, making it effective for real-world applications such as box inpainting.

To summarize, our contributions are as follows:

We propose Constant Acceleration Flow (CAF), a novel ODE framework that integrates acceleration as a controllable variable, enhancing the precision of ODE flow estimation compared to the constant velocity framework.

We propose two strategies to address the flow crossing problem: initial velocity conditioning for the acceleration model and a reflow procedure to improve initial velocity learning. These strategies ensure a more accurate trajectory estimation even in the presence of flow crossings.

Through extensive experiments on synthetic and real datasets, CAF demonstrates remarkable performance, especially achieving the superior FID on CIFAR-10 and Image Net 64 64 over strong baselines. We also demonstrate that CAF learns more accurate flow than Rectified flow by assessing the straightness, coupling preservation, and inversion.

𝜋= 1 𝜋= 0 Generated

2-Rectified Flow CAF (Ours) ℎ= 0 CAF (Ours) ℎ= 1 CAF (Ours) ℎ= 2

Figure 2: 2D synthetic dataset. We compare results between 2-Rectified flow and our Constant Acceleration Flow (CAF) on 2D synthetic data. π0 (blue) and π1 (green) are source and target distributions parameterized by Gaussian mixture models. Here, the number of sampling steps is N = 1. While 2-Rectified flow frequently generates samples that deviate from π1, CAF more accurately estimates the target distribution π1. The generated samples (orange) from CAF form a more similar distribution as the target distribution π1.

2 Related work

Generative models. Learning generative models involves finding a nonlinear transformation between two distributions, typically denoted as π0 and π1, where π0 is a simple distribution like a Gaussian, and π1 is the complex data distribution. Various approaches have been developed to achieve this transformation. For example, variational autoencoders (VAE) [16, 17] optimize the Evidence Lower Bound (ELBO) to learn a nonlinear mapping from the latent space distribution π0 to the data distribution π1. Normalizing flows [18, 19, 20] construct a series of invertible and differentiable mappings to transform π0 into π1. Similarly, GANs [21, 22, 23, 24, 25] earn a generator that transforms π0 into π1 through an adversarial process involving a discriminator. These models typically perform a one-step generation from π0 to π1. In contrast, diffusion models [2, 26, 27, 28, 29, 30] propose learning the probability flow between the two distributions through an iterative process. This iterative process ensures stability and precision, as the model incrementally learns to reverse a diffusion process that adds noise to data. Diffusion models have demonstrated superior performance across various domains, including images [12, 31, 32, 33], 3D [34, 35, 36, 37], and video [38, 39, 40].

Few-step diffusion models Addressing the slow generation speed of diffusion models has become a major focus in recent research: Distillation methods [3, 4, 5, 6, 7, 8, 9] seek to optimize the inference steps of pre-trained diffusion models by amortizing the integration of ODE flow. Consistency models [6, 7, 8] train a model to map any point on the pre-trained diffusion trajectory back to the data distribution, enabling fast generation. Rectified flow [10, 11, 13] is another direction, which focuses on straightening ODE trajectories under a constant velocity field. By straightening the flow and reducing path complexity, it allows for fast generation through efficient and accurate numerical solutions with fewer Euler steps. Recent methods such as AGM [41] also introduce acceleration modeling based on Stochastic Optimal Control (SOC) theory instead of relying solely on velocity. However, AGM predicts time-varying acceleration, which still requires multiple iterative steps to solve the differential equations. In contrast, our proposed CAF ODE assumes that the acceleration term is constant with respect to time. Therefore, there is no need to iteratively solve complex timedependent differential equations. This simplification allows for a direct closed-form solution that supports efficient and accurate sampling in just a few steps.

3 Preliminary

Rectified flow [10, 13] is an ordinary differential equation-based framework for learning a mapping between two distributions π0 and π1. Typically, in image generation, π0 is a simple tractable distribution, e.g., the standard normal distribution, defined in the latent space and π1 is the image distribution. Given empirical observations of x0 π0 and x1 π1 over time t [0, 1], a flow is defined as dxt

dt = v(xt, t), (1)

ℎ= 0 ℎ= 1 ℎ= 2

Sampling direction 𝜋= 1 𝜋= 0

𝑎> 0 𝑎= 0 𝑎< 0

Figure 3: Sampling trajectories of CAF with different h. The sampling trajectories of CAF are displayed for different values of h, which determines the initial velocity and acceleration. π0 and π1 are mixtures of Gaussian distributions. We sample across sampling steps of N = 7 to show how sampling trajectories change with h.

where xt = I(x0, x1, t) is a time-differentiable interpolation between x0 and x1, and v : Rd [0, 1] Rd is a velocity field defined on data-time domain. Rectified flow learns the velocity field v with a neural network vθ by minimizing the following mean square objective:

min θ Ex0,x1 γ,t p(t) h v(xt, t) vθ(xt, t) 2i , (2)

where γ represents a coupling of (π0, π1) and p(t) is a time distribution defined on [0, 1]. The choice of interpolation I leads to various algorithms, such as Rectified flow [10], ADM [30], EDM [29], and LDM [42]. Specifically, Rectified flow proposes a simple linear interpolation between x0 and x1 as xt = (1 t)x0 + tx1, which induces the velocity field v in the direction of (x1 x0), i.e., v(xt, t) = x1 x0. This means the Rectified flow transports x0 to x1 along a straight trajectory with a constant velocity. After training vθ, we can generate a sample x1 using off-the-shelf ODE solvers Φ, such as the Euler method:

xt+ t = xt + t vθ(xt, t), t {0, t, . . . , (N 1) t}, (3)

where t = 1 N and N is the total number of steps. To achieve faster generation with fewer steps without sacrificing accuracy, it is crucial to learn a straight ODE flow. Straight ODE flow minimize numerical errors encountered by the ODE solver.

Reflow and flow crossing. The trajectories of interpolants xt may intersect a phenomenon known as flow crossing due to stochastic coupling between π0 and π1 (e.g., random pairing of x0 and x1). These intersections introduce approximation errors in the neural network, leading to curved sampling trajectories [10]. Our toy experiment, illustrated in Fig. 1a, clearly demonstrates this issue: the simulated sampling trajectories become curved due to flow crossing, rendering one-step simulation inaccurate. To address this problem, Rectified flow [10] introduces a reflow procedure. This procedure iteratively straightens the trajectories by reconstructing a more deterministic and direct pairing of x0 and x1 without altering the marginal distributions. Specifically, the reflow procedure involves generating a new coupling γ of (x0, x1 = Φ(x0; vk θ)) using a pre-trained Rectified flow model vk θ, where k denotes the iteration of the reflow procedure, and Φ(x0; vk θ) = x0 + R 1 0 vk θ(xt, t)dt. By iteratively refining the coupling and the velocity field, the reflow procedure reduces flow crossing, resulting in straighter trajectories and improved accuracy in fewer steps.

We aim to develop a generative model based on the ODE framework that enables faster generation without compromising quality. To achieve this, we propose a novel approach called Constant Acceleration Flow (CAF). Specifically, CAF formulates an ODE trajectory that transports xt with a constant acceleration, offering a more expressive and precise estimation of the ODE flow compared to constant velocity models. Additionally, we propose two novel techniques that address the problem of flow crossing: 1) initial velocity conditioning and 2) reflow procedure for learning initial velocity. The overall training pipeline is presented in Alg. 1.

4.1 Constant Acceleration Flow

We propose a novel ODE framework based on the constant acceleration equation, which is driven by the empirical observations x0 π0 and x1 π1 over time t [0, 1] as:

dxt = v(x0, 0)dt + a(xt, t)tdt, (4)

where v : Rd [0] Rd is the initial velocity field and a : Rd [0, 1] Rd is the acceleration field. We abbreviate time variable t for notation simplicity, i.e., v(x0, 0) = v(x0), a(xt, t) = a(xt). By integrating both sides of (4) with respect to t and assuming a constant acceleration field, i.e., a(xt1) = a(xt2), t1, t2 [0, 1], we derive the following equation:

xt = x0 + v(x0)t + 1

2a(xt)t2. (5)

Given the initial velocity field v, the acceleration field a can be derived as

a(xt) = 2(x1 x0) 2v(x0), (6)

by setting t = 1 and the constant acceleration assumption. Then, we propose a time-differentiable interpolation I as:

xt = I(x0, x1, t, v(x0)) = (1 t2)x0 + t2x1 + v(x0)(t t2), (7)

by substituting (6) to (5). Using this result, we can easily simulate an intermediate sample xt on our CAF ODE trajectory.

Learning initial velocity field. Selecting an appropriate initial velocity field is crucial, as different initial velocities lead to distinct flow dynamics. Here, we define the initial velocity field as a scaled displacement vector between x1 and x0:

v(x0) = h(x1 x0), (8)

where h R is a hyperparameter that adjusts the scale of the initial velocity. This configuration enables straight ODE trajectories between distributions π0 and π1, similar to those in Rectified flow. However, varying h changes the flow characteristics: 1) h = 1 simulates constant velocity flows, 2) h < 1 leads to a model with a positive acceleration, and 3) h > 1 results in a negative acceleration, as illustrated in Fig. 3. Empirically, we observe that the negative acceleration model is more effective for image sampling, possibly due to its ability to finely tune step sizes near data distribution.

The initial velocity field is learned using a neural network vθ, which is optimized by minimizing the distance metric d( , ) between the target and estimated velocities as

min θ Ex0,x1 γ,t p(t),xt I [d(v(x0), vθ(xt))] , (9)

where p(t) is a time distribution defined on [0, 1]. Note that our velocity model learns target initial velocity defined at t = 0. This differs from Rectified flow, which learns target velocity field defined over t [0, 1].

Learning acceleration field. Under the assumption of constant acceleration, the acceleration field is derived from (6) as a(xt) = 2(x1 x0) 2v(x0). (10) We learn the acceleration field using a neural network aϕ by minimizing the distance metric d( , ) as:

min ϕ Ex0,x1 γ,t p(t),xt I [d(a(xt), aϕ(xt))] . (11)

In Sec. C, we theoretically show that CAF ODE preserves the marginal data distribution.

Algorithm 1 Training process of Constant Acceleration Flow

Require: deterministic coupling γ, initial velocity scale h, vθ, aϕ.

1: while not converge do 2: x0, x1 γ, t Unif([0, 1]) 3: v(x0) = h(x1 x0) Target initial velocity 4: xt = I(x0, x1, t, v(x0)) Eq. (7) 5: Lvel = d(v(x0), vθ(xt)) 6: θ θ Lvel update θ using SGD with gradient 7: end while 8: while not converge do 9: x0, x1 γ, t Unif([0, 1]), ˆvθ = vθ(x0) 10: a(xt) = 2(x1 x0) 2ˆvθ Target acceleration 11: xt = I(x0, x1, t, ˆvθ) Eq. (7) 12: Lacc = d(sg[a(xt)], aϕ(xt, ˆvθ)) 13: ϕ ϕ Lacc update ϕ using SGD with gradient 14: end while 15: return vθ, aϕ

4.2 Addressing flow crossing

Rectified flow addresses the issue of flow crossing by a reflow procedure. However, even after the procedure, trajectories may still intersect each other. Such intersections hinder learning straight ODE trajectories, as demonstrated in Fig. 1a. Similarly, our acceleration model also encounters the flow crossing problem. This leads to inaccurate estimation, as the model struggles to predict estimation on these intersections correctly. To further address the flow crossing, we propose two techniques.

Initial velocity conditioning (IVC). We propose conditioning the estimated initial velocity ˆvθ = v(x0) as the input of the acceleration model, i.e., aϕ(xt, ˆvθ). This approach provides the acceleration model with auxiliary information on the flow direction, enhancing its capability to distinguish correct estimations and mitigate ambiguity at the intersections of trajectories, as illustrated in Fig. 1. Our IVC circumvents the non-intersecting condition required in Rectified flow (see Theorem 3.6 in [10]), which is a key assumption for achieving a straight coupling γ. By reducing the ambiguity arising from intersections, CAF can learn straight trajectories with less constrained couplings, which is quantitatively assessed in Tab. 4.

To incorporate IVC into learning the acceleration model, we reformulate (11) as: min ϕ Ex0,x1 γ,t p(t),xt I [d (sg[a(xt)], aϕ(xt, ˆvθ))] . (12)

where sg[ ] indicates stop-gradient operation. Since our velocity model learns to predict the initial velocity (see (9)), we ensure that the model can handle both forward and reverse CAF ODEs, which start from x0 and x1, respectively. Thus, our acceleration model can generalize across different flow directions, enabling inversion as demonstrated in Sec. B.2.

Reflow for initial velocity. It is also important to improve the accuracy of the initial velocity model. Following [10], we address the inaccuracy caused by stochastic pairing of x0 and x1 by employing a pre-trained generative model ψ, which constructs a more deterministic coupling γ of x0 and x1. We subsequently use this new coupling γ to train the initial velocity and acceleration models.

4.3 Sampling

After training the initial velocity and acceleration models, we generate samples using the CAF ODE introduced in (4). The discrete sampling process is given by: xt+ t = xt + t vθ(x0) + t t aϕ(xt, t, vθ(x0)), (13)

where N is the total number of steps, t = 1

N , t = i t, and t = (2i+1)

2 t where i {0, ..., N 1} (See Alg. 2). We adopt t since it empirically improves accuracy, especially in the small N regime. Notably, when N = 1 (one-step generation), t simplifies to 1

2, leading to the closed-form solution in (5). See Alg. 3 for inversion algorithm.

Algorithm 2 Sampling process of Constant Acceleration Flow

Require: velocity model vθ, acceleration model aϕ, sampling steps N, π0.

1: x0 π0 2: ˆvθ vθ(x0) 3: for i = 0 to N 1 do 4: t i N 5: t 2i+1

2N 6: ˆaϕ aϕ(xt, vθ)

N ˆaϕ 8: end for 9: return x1

5 Experiment

We evaluate the proposed Constant Acceleration Flow (CAF) across various scenarios, including both synthetic and real-world datasets. In Sec. 5.1, our investigation begins with a simple twodimensional synthetic dataset, where we compare the performance of Rectified flow and CAF to clearly demonstrate the effectiveness of our model. Next, we extend our experiments to real-world image datasets, specifically CIFAR-10 (32 32) and Image Net (64 64), in Sec. 5.2. These experiments highlight CAF s ability to generate high-quality images with a single sampling step. Furthermore, we conduct an in-depth analysis of CAF through evaluations of coupling preservation, straightness, inversion tasks, and an ablation study in Sec. 5.3.

5.1 Synthetic experiments

We demonstrate the advantages of the Constant Acceleration Flow (CAF) over the constant velocity flow model, Rectified Flow [10], through synthetic experiments. For the neural networks, we use multilayer perceptrons (MLPs) with five hidden layers and 128 units per layer. Initially, we train 1-Rectified flow on 2D synthetic data to establish a deterministic coupling. We then train both CAF and 2-Rectified flow. For CAF, we incorporate the initial velocity into the acceleration model by concatenating it with the input, ensuring that the model capacities of both CAF and 2-Rectified flow remain comparable. We set d as l2 distance. Fig. 2 presents samples generated from CAF in one step and from 2-Rectified flow in two steps. Our CAF more accurately approximates the target distribution π1 than 2-Rectified flow. In particular, CAF with h = 2 (negative acceleration) learns the most accurate distribution. In contrast, 2-Rectified flow frequently generates samples that significantly deviate from π1, indicating its difficulty in accurately estimating straight ODE trajectories. This experiment shows that reflowing alone may not overcome the flow crossing problem, leading to poor estimations, whereas our proposed acceleration modeling and IVC effectively address this issue. Moreover, Fig. 3 shows sampling trajectories from CAF trained with different hyperparameters h. It clearly demonstrates that h controls the flow dynamics as we intended: h > 1 indicates negative acceleration, h = 1 represents constant velocity, and h < 1 corresponds to positive acceleration flows. Additional synthetic examples are provided in Fig. 6.

5.2 Real-data experiments

To further validate the effectiveness of our approach, we train CAF on real-world image datasets, specifically CIFAR-10 at 32 32 resolution and Image Net at 64 64 resolution. To create a deterministic coupling γ, we utilize the pre-trained EDM models [29] and adopt the U-Net architecture of ADM [30] for the initial velocity and acceleration models. In the acceleration model, we double the input dimension of first layer to concatenate the initial velocity to the input xt of the acceleration model, which marginally increases the total number of parameters. We set h = 1.5 and d as LPIPS-Huber loss [43] for all real-data experiments.

Baselines and evaluation. We evaluate state-of-the-art diffusion models [1, 2, 7, 28, 29], GANs [22, 23, 24], and few-step generation approaches [6, 7]. We primarily assess the image generation quality of our method using the Fréchet Inception Distance (FID) [50] and Inception Score (IS) [51]. Additionally, we evaluate diversity using the recall metric following [6, 7, 10].

Table 1: Performance on CIFAR-10.

Model N Unconditional Conditional

FID FID GAN Models

Big GAN [22] 1 8.51 - Style GAN-Ada [23] 1 2.92 2.42 Style GAN-XL [24] 1 - 1.85 Diffusion/Consistency Models

Score SDE [1] 2000 2.20 - DDPM [2] 1000 3.17 - VDM [27] 1000 7.41 - LSGM [28] 138 2.10 - DDIM [26] 10 13.36 -

EDM [29] 35 2.01 1.82 5 37.75 35.54

CT [6] 2 5.83 - 1 8.70 - Diffusion/Consistency Models Distillation

Diff-Instruct [9] 1 4.53 - DMD [44] 1 3.77 - DFNO [5] 1 3.78 - TRACT [45] 1 3.78 - KD [46] 1 9.36 -

CD [6] 2 2.93 - 1 3.55 -

CTM [7] 2 1.87 1.63 1 1.98 1.73 Rectified Flow Models

2-Rectified Flow [10] 2 7.89 3.74 1 11.81 6.88 2-Rectified Flow + Distill [10] 1 4.84 -

CAF (Ours) 1 4.81 2.68 CAF + GAN (Ours) 1 1.48 1.39

Table 2: Performance on Image Net 64 64.

Model N FID IS Rec

Big GAN-deep [22] 1 4.06 - 0.48 Style GAN-XL [24] 1 2.09 82.35 0.52 Diffusion/Consistency Models

DDIM [26] 50 13.7 - 0.56 10 18.3 - 0.49 DDPM [2] 250 11.0 - 0.58 i DDPM [47] 250 2.92 - 0.62 ADM [30] 250 2.07 - 0.63

EDM [29] 79 2.44 48.88 0.67 5 55.3 - -

DPM-solver [48] 20 3.42 - - 10 7.93 - -

DEIS [49] 20 3.10 - - 10 6.65 - -

CT [6] 2 11.1 - 0.56 1 13.0 - 0.47 Diffusion/Consistency Models Distillation

Diff-Instruct [9] 1 5.57 - - DMD [44] 1 2.62 - - TRACT [45] 1 7.43 - - DFNO [5] 1 7.83 - 0.61 PD [3] 1 15.39 - 0.62

CD [6] 2 4.70 - 0.64 1 6.20 40.08 0.57

CTM [7] 2 1.73 64.29 0.57 1 1.92 70.38 0.57 Rectified Flow Models

CAF (Ours) 1 6.52 37.45 0.62 CAF + GAN (Ours) 1 1.69 62.03 0.64

Distillation. Distilling a few-step student model from a pre-trained teacher model has recently become essential for high-quality few-step generation [6, 7, 10, 11]. Insta Flow [11] has observed that learning straighter trajectories and achieving good coupling significantly enhance distillation performance. Moreover, CTM [7] and DMD [44] incorporate an adversarial loss as an auxiliary loss to facilitate the training of the student model. We empirically found that incorporating the adversarial loss alone was sufficient to achieve superior performance for one-step sampling without introducing instability. For training details, please refer to Sec. A. CIFAR-10. We present the experimental results on CIFAR-10 in Tab. 1. Our base unconditional CAF model (4.81 FID, N = 1) significantly improves the FID compared to recent state-of-the-art diffusion models (without distillation), including DDIM [26] (13.36 FID, N = 10), EDM (37.75 FID, N = 5), and 2-Rectified flow (7.89 FID, N = 2) in a few-step generation (e.g., N < 10). We retrained 2-Rectified flow using the official codes of [10], achieving a slightly better performance than the officially reported performance (12.21 FID) for one-step generation [10]. CAF s remarkable 3.08 FID improvement over 2-Rectified flow (N = 2) highlights the effectiveness of acceleration modeling in a fast generation. Our approach is also effective in class-conditional generation, where the base CAF model (2.68 FID, N = 1) shows a significant FID improvement over EDM (35.54 FID, N = 5) and 2-Rectified flow (3.74 FID, N = 2). Additionally, after adversarial training, CAF achieves a superior FID of 1.48 for unconditional generation and 1.39 for conditional generation with N = 1. Lastly, we qualitatively compare the 2-Rectified flow and our CAF in Fig. 4, where CAF generates more vivid samples with intricate details than 2-Rectified flow. Image Net. We extend our evaluation to the Image Net dataset at 64 64 resolution to demonstrate the scalability and effectiveness of our CAF model on more complex and higher-resolution images. Similar to the results on CIFAR-10, our base conditional CAF model significantly improves the FID compared to recent state-of-the-art diffusion models (without distillation) in the small N regime (e.g., N < 10). Specifically, CAF (6.52 FID, N = 1) outperforms models such as DPM-solver [48] (7.93 FID, N = 10), CT [6] (11.1 FID, N = 2), and EDM [29] (55.3 FID, N = 5). This validates that the superior performance of CAF can be effectively generalized to complex and large-scale datasets. Additionally, after adversarial training, CAF outperforms or is competitive with state-of-the-art distillation baselines in one-step generation. Notably, CAF achieves the best FID performance of 1.69, surpassing strong baselines. We also demonstrate one-step qualitative results in Fig. 14.

1 step 10 steps

CAF (Ours) + Distilled

2-RF + Distlled

CAF (Ours) + Distilled

2-RF + Distlled

Figure 4: Qualitative results on CIFAR-10. We compare the quality of generated images from 2-Rectified flow and CAF (Ours) with N = 1 and 10. Each image x1 is generated from the same x0 for both models. CAF generates more vivid images with intricate details than 2-RF for both N.

Table 3: Coupling preservation.

Metric 2-Rectified Flow CAF (ours)

LPIPS 0.092 0.041 PSNR 29.79 33.16

Table 4: Flow straightness comparison.

Dataset 2-Rectified Flow CAF (ours)

2D 0.065 0.058 CIFAR-10 0.043 0.034

Table 5: Ablation study on CIFAR-10 (N = 1).

Config Constant acceleration v0 condition Reflow procedure FID

A 378 B 6.88 C (h=1.5) 3.82 D (h=1.5) 2.68 E (h=1) 3.02 F (h=0.5) 2.73

5.3 Analysis

Coupling preservation. We evaluate how accurately CAF and Rectified flow approximate the deterministic coupling obtained from pre-trained models via a reflow procedure. To analyze this, we first conduct synthetic experiments where the interpolant paths I are crossed, as illustrated in Fig. 5a. Due to the flow crossing, the sampling trajectory of Rectified flow fails to preserve the ground-truth coupling (interpolation path I), leading to a curved sampling trajectory. In contrast, our CAF learns the straight interpolation paths by incorporating acceleration, demonstrating superior coupling preservation ability.

Moreover, we evaluate the coupling preservation ability on real data from CIFAR-10. We randomly sample 1K training pairs (x0, x1) from the deterministic coupling γ and measure the similarity between x1 and ˆx1, where ˆx1 is a generated sample from x0. In other words, we measure the distance between a ground truth image and a generated image corresponding to the same noise. If the coupling is well-preserved, the distance should be small. We use PSNR and LPIPS [52] as distance measures. The result in Tab. 3 demonstrates that CAF better preserves coupling. In terms of PSNR, CAF outperforms Rectified flow by 3.37. This is consistent with the qualitative result in Fig. 5b, where ˆx1 from CAF resembles more to x1 (ground truth) than ˆx1 from Rectified flow.

Flow straightness. To evaluate the straightness of learned trajectories, we introduce the Normalized Flow Straightness Score (NFSS). Similar to previous works [10, 11], we measure flow straightness S by the L2distance between the normalized displacement vector (x0 x1) and the normalized velocity vector xt as below:

S = Ex0,x1,t

" x1 x0 x1 x0 2 xt xt 2

Here, a smaller value of S indicates a straighter trajectory. We compare S between CAF and Rectified flow using synthetic and real-world datasets, as presented in Tab. 4. For Rectified flow, we use xt = vθ(xt), while for CAF, we use xt = vθ(x0) + aϕ(xt)t. The results show that CAF outperforms Rectified flow in flow straightness.

𝜋= 1 𝜋= 0 Interpolation path Sampling trajectory

RF CAF (Ours)

2-RF (0.250)

CAF (0.081) GT

2-RF (0.181)

CAF (0.048) GT

2-RF (0.328)

CAF (0.057) GT 2-RF (0.302)

CAF (0.023) GT

2-RF (0.343)

CAF (0.048) GT 2-RF (0.265)

CAF (0.050) GT

Figure 5: Experiments for coupling preservation. (a) We plot the sampling trajectories during training where their interpolation paths I are crossed. Due to the flow crossing, RF (top) rewires the coupling, whereas CAF (bottom) preserves the coupling of training data. (b) CAF accurately generates target images from the given noise (e.g., a car from the car noise), while RF often fails (e.g., a frog from the car noise). LPIPS [52] values are in parentheses.

Inversion We further demonstrate CAF s capability in real-world applications by conducting zeroshot tasks such as reconstruction and box inpainting using inversion. We provide implemenetation details and algorithms in Sec. B.2. As shown in the Tab. 6 and 7, our method achieves lower reconstruction errors (CAF: 46.68 PSNR vs. RF: 33.34 PSNR) and better zero-shot inpainting capabilities even with fewer steps compared to baselines. These improvements are attributed to CAF s superior coupling preservation capability. Moreover, we present qualitative comparisons between CAF and the baselines in Fig. 12 and 13, which further validates the quantitative results.

Ablation study. We conduct an ablation study to evaluate the effectiveness of components in our framework under the one-step generation setting (N = 1). We examine the improvements achieved by 1) constant acceleration modeling, 2) initial velocity (v0) conditioning, and 3) the reflow procedure for v0. The configurations and results are outlined in Tab. 5. Specifically, A and B correspond to 1-Rectified flow and 2-Rectified flow, respectively. Configurations C to F represent our CAF frameworks, with C being our CAF without IVC. By comparing A,B,C, and F, we demonstrate that all three components in our framework substantially improve the performance. In addition, we analyze the final model across various acceleration scales controlled by h. The performance difference between D and F is relatively small, indicating that our framework is robust to the choice of hyperparameters. Empirically, we observe that configuration F, i.e., CAF (h = 1.5) with negative acceleration, achieves the best FID of 2.68. Notably, our CAF without v0 conditioning, still outperforms rectified flow (configuration B) by 3.06 FID. This highlights the critical role of constant acceleration modeling in enhancing the quality of few-step generation. Also, we verify the significance of reflowing by comparing configurations A and B, which achieve 378 FID and 6.88 FID, respectively.

6 Conclusion

In this paper, we have introduced the Constant Acceleration Flow (CAF) framework, which enhances precise ODE trajectory estimation by incorporating a controllable acceleration variable into the ODE framework. To address the flow crossing problem, we proposed two strategies: initial velocity conditioning and a reflow procedure. Our experiments on toy datasets, real-world dataset demonstrate CAF s capabilities and scalability, achieving state-of-the-art FID scores. Furthermore, we conducted extensive ablation studies and analyses including assessments of flow straightness, coupling preservation, and real-world applications to validate and deepen our understanding of the effectiveness of our proposed components in learning accurate ODE trajectories. We believe that CAF offers a promising direction for efficient and accurate generative modeling, and we look forward to exploring its applications in more diverse settings such as 3D and video.

Acknowledgement

This work was supported by ICT Creative Consilience Program through the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (IITP-2024-RS-2020-II201819, 10%), the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2023R1A2C2005373, 45%), and the Virtual Engineering Platform Project (Grant No. P0022336, 45%), funded by the Ministry of Trade, Industry & Energy (Mo TIE, South Korea).

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A Implementation details

We utilize the pre-trained EDM model [29] to build the deterministic coupling γ for training our models. To construct deterministic couplings for CIFAR-10 and Image Net, we select N = 18 and N = 40, respectively, using deterministic sampling following the protocol in [29]. For CIFAR-10 and Image Net, we generate 1M and 3M pairs, respectively. We use the batch size of 2048 and train for 700K/700K iterations on Image Net. For CIFAR-10, we use the batch size of 512 and train for 500K/500K iterations. For all experiments, we use Adam W [53] optimizer with a learning rate of 0.0001 and apply an Exponential Moving Average (EMA) with a 0.999 decay rate. For training acceleration model, we initialize it with initial velocity model for faster convergence.

For adversarial training, we employ adversarial loss Lgan using real data x1,real from [24]:

Lgan,η(ϕ) = Ex1,real [log dη(x1,real)] + Ex0 [log(1 dη(ˆx1))] , (15)

where dη is a discriminator and ˆx1 = x0 + vθ(x0) + 1

2aϕ(x0, vθ(x0)). In the end, we use the following combined loss to update the acceleration model:

L(ϕ, η) = Lacc(ϕ) + λgan Lgan(ϕ, η), (16)

where Lacc corresponds to (12) and λ are weight hyperparameters. Following [42, 54], we employ adaptive weighting as λgan = ϕl Lacc(ϕ) ϕl Lgan(ϕ,η) , where ϕl is the last layer of the acceleration model. Without Lacc, we found the training unstable and frequently exhibit mode collapse issue, which is a common problem with adversarial training. We follow the training configuration from Style GANXL [24]. We bilinearly upscale the image to 224 224 resolution and use Efficient Net [55] and Dei Tbase [56] for extracting features. During the adversarial training, we only optimize the acceleration model and discriminator with the learning rate of 2e-5 and 1e-3, respectively. We keep the parameters of the initial velocity model fixed for stable training. The total training takes about 21 days with 8 NVIDIA A100 GPUs for Image Net, and takes 10 days 8 NVIDIA RTX3090 GPUs for CIFAR-10.

B Additional results

B.1 Additional qualitative results

2D toy dataset. In Fig. 6, we provide additional generation results and sampling trajectories on various 2D synthetic datasets with N = 1, demonstrating the effectiveness of our approach for fast generation. Fig. 7 provides additional examples of coupling preservation on 2-RF and CAF.

Real-world dataset. In Fig. 8 and 9, we show additional generation results from our base CAF model on CIFAR-10 with N = 1, 10, and 50. In Fig. 10, we compare the generation result between 2-RF and CAF distilled versions. Fig. 11 shows sampling results from our base CAF models with different hyperparameters h. Lastly, Fig. 14 shows the generation results on Image Net with N = 1.

B.2 Real-world applications

Inversion techniques are essential for real-world applications such as image and video editing [57, 58]. However, existing methods typically require 25 100 steps for accurate inversion, which can be computationally intensive. In contrast, our method significantly reduces the inference time by enabling inversion in just a few steps (e.g., N < 20). We demonstrate this efficiency in two tasks: reconstruction and box inpainting.

To reconstruct x1, we first invert x1 to obtain ˆx0, as described in Alg. 3. We then use the generation process (Alg. 2) with ˆx0 and same initial velocity vθ(x1) used in Alg. 3 to generate ˆx1. For box inpainting, we inject conditional information the non-masked image region into the iterative inversion and generation procedures, as detailed in Alg. 4. As demonstrated in Tab. 6 and 7, our method achieves better reconstruction quality (CAF: 46.68 PSNR vs. RF: 33.34 PSNR) and zeroshot inpainting capability even with fewer steps compared to baseline methods. Qualitative results are presented in Fig. 12 and 13, which further illustrate the effectiveness of our approach. This demonstrate that our method can be efficiently used for real-world applications, offering both speed and accuracy advantages over existing techniques.

Algorithm 3 Inversion process of Constant Acceleration Flow

Require: velocity model vθ, acceleration model aϕ, sampling steps N, π1.

1: x1 π1 2: ˆvθ vθ(x1) 3: for i = N to 1 do 4: t i N 5: t 2i 1

2N 6: ˆaϕ aϕ(xt, ˆvθ)

N ˆaϕ 8: end for 9: return x0

Algorithm 4 Box inpainting of Constant Acceleration Flow

Require: velocity model vθ, acceleration model aϕ, sampling steps N, reference image x1, binary image mask Ωwhere 1 indicates the missing pixels. 1: σ N(0, I) 2: x x1 (1 Ω) + σ Ω Create image with missing pixels and add noise σ 3: ˆvθ vθ( x) 4: for i = N to 1 do Inversion steps 5: t i N , t 2i 1

2N 6: ˆaϕ aϕ(xt, ˆvθ)

N ˆaϕ 8: xt 1

N (1 Ω) + (1 t)σ Ω, σ N(0, I) 9: end for 10: ˆvθ vθ(x0) 11: for j = 0 to N 1 do Generation steps 12: t j N , t 2j+1

2N 13: ˆaϕ aϕ(xt, ˆvθ)

N ˆaϕ 15: xt+ 1

N x1 (1 Ω) + xt+ 1

N Ω 16: end for 17: return inpainted image x1

B.3 Comparison with previous acceleration modeling literatures

Here, we elaborate on the crucial differences between AGM [41] and CAF. The main distinction is that CAF assumes constant acceleration, whereas AGM predicts time-dependent acceleration. Since the CAF ODE assumes that the acceleration term is constant with time, there is no need to solve time-dependent differential equations iteratively. This allows for a closed-form solution that supports efficient and accurate sampling, given that the learned velocity and acceleration models are accurate. Specifically, the solution for CAF ODE is given by:

x1 = x0 + Z 1

0 v(x0) + a(xt) tdt = x0 + v(x0) + Z 1

0 a(xt) tdt (17)

= x0 + v(x0) + a(xt) Z 1

0 tdt = x0 + v(x0) + 1

2a(xt) (18)

The integral simplifies thanks to the constant acceleration assumption, leading to one-step sampling. In contrast, AGM s acceleration is time-varying, meaning that the differential equation cannot be reduced in an analytic form. It requires multiple steps to approximate the true solution accurately. In Tab. 8, we systemically compare AGM with our CAF, where CAF consistently outperforms AGM. Moreover, we conducted additional experiments where AGM was trained with deterministic couplings as in our reflow setting. Incorporating reflow into AGM did not improve its performance in the few-step regime, which further highlights the distinct advantage of CAF over AGM.

Table 6: Reconstruction error.

Model N PSNR LPIPS

CM - N/A N/A CTM - N/A N/A EDM 4 13.85 0.447 2-RF 2 33.34 0.094 2-RF 1 29.33 0.204

CAF (Ours) 1 46.68 0.007 CAF (+GAN) (Ours) 1 40.84 0.028

Table 7: Box inpainting.

Model NFE FID

CM 18 13.16 CTM - N/A EDM - N/A 2-RF 12 16.41

CAF (Ours) 12 10.39 CAF (+GAN) (Ours) 12 10.91

Table 8: Comparison between AGM and CAF.

Model Acceleration Closed-form solution Reflow for velocity FID on CIFAR-10

AGM [41] Time-varying No No 11.88 (N = 5) AGM (enhanced ver.) Time-varying No Yes 15.23 (N = 5)

CAF (Ours) Constant Yes Yes 4.81 (N = 1)

C Marginal preserving property of Constant Acceleration Flow

We demonstrate that the flow generated by our Constant Acceleration Flow (CAF) ordinary differential equation (ODE) maintains the marginal of the data distribution, as established by the definitions and theorem in [10]. Definition C.1. For a path-wise continuously differentiable process x = {xt : t [0, 1]}, we define its expected velocity vx and acceleration ax as follow:

vx(x, t) = E dxt

dt | xt = x , ax(x, t) = E d2xt

dt2 | xt = x , x supp(xt). (19)

For x / supp(xt), the conditional expectation is not defined and we set vx and ax arbitrarily, for example vx(x, t) = 0 and ax(x, t) = 0. Definition C.2. [10] We denote that x is rectifiable if vx is locally bounded and the solution to the integral equation of the form

zt = z0 + Z t

0 vx(zt, t)dt, t [0, 1], z0 = x0, (20)

exists and is unique. In this case, z = {zt : t [0, 1]} is called the rectified flow induced by x. Theorem 1. [10] Assume x is rectifiable and z is its rectified flow. Then Law(zt) = Law(xt), t [0, 1].

Refer to [10] for the proof of Theorem 1.

We will now show that our CAF ODE satisfies Theorem 1 by proving that our proposed ODE (4) induces z, which is the rectified flow as defined in Definition C.2. In (4), we define the CAF ODE as

dt2 t. (21)

By taking the conditional expectation on both sides, we obtain

vx(x, t) = vx(x, 0) + ax(x, t) t, (22)

from Definition C.1. Then, the solution of the integral equation of CAF ODE is identical to the solution in Definition C.2 by (22):

zt = z0 + Z t

0 vx(z0, 0) + ax(zt, t) tdt (23)

0 vx(zt, t)dt. (24)

This indicates that z induced by CAF ODE is also a rectified flow. Therefore, the CAF ODE satisfies the marginal preserving property, i.e., Law(zt) = Law(xt), as stated in Theorem 1.

D Limitation and Broader impacts

D.1 Limitations

One limitation of our model is the increased number of function evaluations (NFE) required for N-step generation. While Rectified flow achieves an NFE of N by only computing the velocity at each step, our method necessitates an additional computation, resulting in a total NFE of N + 1. This is because we compute the initial velocity at the beginning and the acceleration at each step. Although this extra evaluation slightly increases the computational burden, it is relatively minor in terms of overall performance and still enables efficient few-step generation. Moreover, this additional step can be reduced by jointly predicting velocity and acceleration terms with a single model, which we leave for future work. Another limitation is the additional effort required to generate supplementary data. We utilize generated data to create a deterministic coupling of noise and data samples for training CAF. While generating more data enhances our model s performance, it can increase GPU usage, leading to higher carbon emissions.

D.2 Broader Impacts

Recent advancements in generative models hold significant potential for societal benefits across a wide array of applications, such as image and video generation and editing, medical imaging analysis, molecular design, and audio synthesis. Our CAF framework contributes to enhancing the efficiency and performance of existing diffusion models, offering promising directions for positive impacts across multiple domains. This suggests that in practical applications, users can utilize generative models more rapidly and accurately, enabling a broad spectrum of activities. However, it is crucial to acknowledge potential risks that must be carefully managed. The increased accessibility of generative models also broadens the potential for misuse. As these technologies become more widespread, the possibility of their exploitation for fraudulent activities, privacy breaches, and criminal behavior increases. It is vital to ensure their ethical and responsible use to prevent negative impacts. Establishing regulated ethical standards for developing and deploying generative AI technologies is necessary to prevent such misuse. Additionally, imposing restricted access protocols or verification systems to trace and authenticate generated contents will help ensure their responsible use.

𝜋= 1 𝜋= 0 Generated

2-Rectified Flow CAF (Ours) ℎ= 0 CAF (Ours) ℎ= 1 CAF (Ours) ℎ= 2

(a) Generation results

Sampling direction 𝜋= 1 𝜋= 0

ℎ= 0 ℎ= 1 ℎ= 2

(b) Sampling trajectories with different h

𝜋= 1 𝜋= 0 Generated

2-Rectified Flow CAF (Ours) ℎ= 0 CAF (Ours) ℎ= 1 CAF (Ours) ℎ= 2

(c) Generation results

Sampling direction 𝜋= 1 𝜋= 0

ℎ= 0 ℎ= 1 ℎ= 2

(d) Sampling trajectories with different h

Figure 6: Experiments on various 2D synthetic dataset. We compare results between 2-Rectified Flow and our Constant Acceleration Flow (CAF) on 2D synthetic data. π0 (blue) and π1 (green) are source and target distributions parameterized by Gaussian mixture models. The generated samples (orange) from CAF form a more similar distribution as the target distribution π1.

Figure 7: Additional visualizations of coupling preservation on CIFAR-10. CAF accurately generates target images (x1) from the given noise (x0), while Rectified Flow often fails to preserve coupling of x0 and x1 .

Figure 8: Qualitative results on unconditional generation (CIFAR-10). We illustrate generating images with varying sampling steps, demonstrating consistency quality even for a one-step generation.

airplane automobile bird cat deer dog frog horse ship truck

Figure 9: Qualitative results on conditional generation (CIFAR-10). We illustrate generating images with varying sampling steps, demonstrating consistency quality even for a one-step generation.

Figure 10: Comparisons on unconditional generation (CIFAR-10). We compare distilled model from 2-Rectified Flow (2-RF+Distill+GAN) and CAF (CAF+Distill+GAN) with qualitative results.

1 step h = 1

10 steps h = 1

Figure 11: Unconditional generation for different h on CIFAR-10. We display qualitative results of CAF for different values of h, indicating that our framework is robust to the choice of h.

(a) Ground Truth

(b) CAF (Ours) (1 step, PSNR=46.68, LPIPS=0.007)

(c) RF (1 step, PSNR=29.33, LPIPS=0.204)

Figure 12: Reconstruction results using inversion.

(a) Masked Images

(b) CAF (ours) (12 step, FID=10.39)

(c) 2-RF (12 step, FID=16.41)

(d) CM (18 step, FID=13.16)

Figure 13: Zero-shot box inpainting results. We use a 16 16 size mask for masked images in (a). For consistency model in (d), we followed their official code for inpainting.

Figure 14: Qualitative results on conditional generation for Image Net 64 64 (N = 1, FID=1.69).

Neur IPS Paper Checklist

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Answer: [Yes]

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Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We have discussed the limitations of our work in the supplementary material.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. The authors are encouraged to create a separate "Limitations" section in their paper. The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren t acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: We have provided complete proof in the supplementary materials. Guidelines:

The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. All assumptions should be clearly stated or referenced in the statement of any theorems. The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. Theorems and Lemmas that the proof relies upon should be properly referenced. 4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: We have provided implementation details in the supplementary materials. Guidelines:

The answer NA means that the paper does not include experiments. If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. While Neur IPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. (b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes] Justification: We will provide an official code. Guidelines:

The answer NA means that paper does not include experiments requiring code. Please see the Neur IPS code and data submission guidelines (https://nips.cc/ public/guides/Code Submission Policy) for more details. While we encourage the release of code and data, we understand that this might not be possible, so No is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). The instructions should contain the exact command and environment needed to run to reproduce the results. See the Neur IPS code and data submission guidelines (https: //nips.cc/public/guides/Code Submission Policy) for more details. The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: We have provided details in the supplementary materials. Guidelines:

The answer NA means that the paper does not include experiments. The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. The full details can be provided either with the code, in appendix, or as supplemental material. 7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [No] Justification: Single run. Guidelines:

The answer NA means that the paper does not include experiments. The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: We have provided details in the supplementary materials. Guidelines:

The answer NA means that the paper does not include experiments. The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper). 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines? Answer: [Yes] Justification: We have followed the code of ethics. Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: We have discussed the broader impact in supplementary materials. Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to

generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [Yes] Justification: We will provide code and checkpoint with the safeguards. Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: We have credited and respected the assets in the paper. Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA] Justification: We have not released new asset. Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justification: We have not done any crowdsourcing experiments. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justification: We have not done any crowdsourcing experiments. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.