Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow

## Abstract

Abstract Diffusion models have greatly improved visual generation but are hindered by slow generation speed due to the computationally intensive nature of solving generative ODEs. Rectified flow, a widely recognized solution, improves generation speed by straightening the ODE path. Its
key components include: 1) using the diffusion form of flow-matching, 2) employing 𝒗 𝒗 \bm{v} bold_italic_v -prediction, and 3) performing rectification (a.k.a. reflow). In this paper, we argue that the success of rectification primarily lies in using a pretrained diffusion model to obtain matched pairs of noise and samples, followed by retraining with these matched noise-sample pairs. Based on this, components 1) and 2) are unnecessary. Furthermore, we highlight that straightness is not an essential training target for rectification; rather, it is a specific case of flow-matching models. The more critical training target is to achieve a first-order approximate ODE path, which is inherently curved for models like DDPM and Sub-VP. Building on this insight, we propose Rectified Diffusion, which generalizes the design space and application scope of rectification to encompass the broader category of diffusion models, rather than being restricted to flow-matching models. We validate our method on Stable Diffusion v1-5 and Stable Diffusion XL. Our method not only greatly simplifies the training procedure of rectified flow-based previous works (e.g., InstaFlow) but also achieves superior performance with even lower training cost. Our code is available at https://github.com/G-U-N/Rectified-Diffusion .

## 1 Introduction

Diffusion models have greatly advanced the field of visual generation, enabling the creation of high-quality images and vivid videos from text (Ho et al., ; Song et al., ; Rombach et al., ; Singer et al., ; Podell et al., ; Esser et al., ; Shi et al., ). However, the generation process of diffusion models involves solving an expensive generative ODE numerically, which significantly slows down the generation speed compared to other generative models (e.g., GAN) (Goodfellow et al., ; Sauer et al., ; ). A widely recognized solution to this issue is rectified flow. The training target of rectified flow, as highlighted in the previous works (Liu et al., ; ; Yan et al., ), is to make the new ODE path straighter, enabling the models to generate high-fidelity images with fewer steps while retaining the flexibility of sampling with more inference steps for further quality enhancement. The key components of rectified flow are threefold:

- 1)
Flow-Matching. Rectified flow proposes to employ the flow-matching based diffusion form (Liu et al., ; Lipman et al., ). The intermediate noisy state $\mathbf{x}_{t}$ is defined as $(1-t)\mathbf{x}_{0}+t\bm{\epsilon}$, where $\mathbf{x}_{0}$ is the clean data, $\bm{\epsilon}\sim\mathcal{N}(\bm{0},\mathbf{I})$ is normal noise, and $t\in[0,1]$ is the timestep. This design is more straightforward compared to the semi-linear form of the original DDPM (Ho et al., ).
- 2)
$\bm{v}$-prediction. Rectified flow proposes to adopt $\bm{v}$-prediction (Salimans & Ho, ; Liu et al., ). That is, the model learns to predict $\bm{v}=\mathbf{x}_{0}-\bm{\epsilon}$. This makes the denoising form simple. For example, one can predict $\mathbf{x}_{0}$ based on $\mathbf{x}_{t}$ with $\hat{\mathbf{x}}_{0}=\mathbf{x}_{t}+t\bm{\hat{}}{v}_{\bm{\theta}}$, where $\bm{\theta}$ denotes the model parameters and $\hat{\phantom{x}}$ denotes the predictions. Moreover, it avoids the numerical issue when $t\approx 1$ with $\bm{\epsilon}$-prediction. For example, $\hat{\mathbf{x}}_{0}=\frac{\mathbf{x}_{t}-t\hat{\bm{\epsilon}}_{\bm{\theta}}}{
1-t}\approx\frac{\mathbf{x}_{t}-t\hat{\bm{\epsilon}}_{\bm{\theta}}}{0}$, which is invalid.
- 3)
Rectification. Rectification, also known as reflow, is an important technique proposed in rectified flow (Liu et al., ). It is a progressive retraining technique that greatly improves the generation quality at low-step regime and maintains the flexibility of standard diffusion models. To be specific, it turns an arbitrary coupling of $\mathbf{x}_{0}\sim\mathbb{P}_{0}$ (real data) and $\bm{\epsilon}\sim\mathbb{P}_{1}$ (noise) adopted in standard diffusion training to a new deterministic coupling of $\hat{\mathbf{x}}_{0}\sim\mathbb{P}^{\bm{\theta}}_{0}$ (generated data) and $\bm{\epsilon}\sim\mathbb{P}_{1}$ (pre-collected noise). To put it in a nutshell, it replaces the $\mathbf{x}_{t}=(1-t)\mathbf{x}_{0}+t\bm{\epsilon}$ with $\mathbf{x}_{t}=(1-t)\hat{\mathbf{x}}_{0}+t\hat{\bm{\epsilon}}$, where $\mathbf{x}_{0}$ is real data, $\hat{\mathbf{x}}_{0}$ is data generated by pretrained diffusion models $\bm{\theta}$, $\bm{\epsilon}$ is the randomly sampled noise, and $\hat{\bm{\epsilon}}$ is the noise used to generate data $\hat{\mathbf{x}}_{0}$. Previous works emphasize the rectification procedure is only feasible to $\bm{v}$-prediction based flow-matching models. That is, they believe the first two points are the foundations to adopt rectification for improving efficiency. And they emphasize the rectification procedure ‘straightens’ the ODE path.

Figure: Figure 1: Overview of comparison between Rectified Flow and Our Rectified Diffusion.
Refer to caption: x1.png

The motivation of this paper is to investigate what is most essential about rectified flow. We argue that the effectiveness of rectified flow stems from using a pretrained diffusion model to acquire matched pairs of noise and samples, followed by retraining with these matched noise-sample pairs (i.e., the aforementioned third point). Based on this, the aforementioned first two points (i.e., flow-matching & $\bm{v}$-prediction) are unnecessary. This allow us generalize the design space of rectified flow and make it adoptable for different diffusion variants including DDPM (Ho et al., ), EDM (Karras et al., ), Sub-VP (Song et al., ), and etc.

To this end, we propose Rectified Diffusion, as illustrated in Fig. 1, our overall design is straightforward. We keep everything of the pretrained diffusion models unchanged, including noise schedulers, prediction types, networks architectures, and even training and inference code. The only difference is that the noise $\bm{\epsilon}$ and data $\mathbf{x}_{0}$ adopted for training are pre-collected and generated by the pretrained diffusion models instead of independently sampled from Gaussian and real data datasets.

Additionally, we highlight that straightness is no more an essential training target when we generalize the design space from solely flow-matching to more general diffusion forms. We analyze and show that the training target of Rectified Diffusion is to obtain a first-order approximate ODE path. In simple terms, a first-order ODE implies the predictions of models remain consistent along the ODE trajectory and it still maintains at the same ODE trajectory after each denoising step. For models like DDPM (Ho et al., ), the first-order ODE path is inherently curved instead of straight. Therefore, ‘straightness’ is no more suitable for Rectified Diffusion and is just a special case when we use the form of flow-matching.

To empirically validate our claim, we conduct experiments using Stable Diffusion, comparing our approach with InstaFlow (Liu et al., ), a key baseline based on rectified flow for text-to-image generation. We adhere the training setting of InstaFlow. The primary distinction is that InstaFlow requires transforming the Stable Diffusion models into a $\bm{v}$-prediction flow-matching model, while our method leaves everything of original Stable Diffusion unchanged. Our results demonstrate apparently better performance and faster training, likely due to our minimal differences in diffusion configurations. Our one-step performance achieves significantly superior performance with only 8% trained images of InstaFlow as shown in Fig. 2.

Besides, we propose to replace the second-stage distillation adopted in InstaFlow with consistency distillation. We observe that the first-order approximate ODE path greatly facilitates consistency distillation, allowing us to achieve better performance at 3% the GPU days than the further distilled model of InstaFlow. Additionally, we introduce Rectified Diffusion (Phased), which divides the ODE path along the time axis into multiple segments and enforces first-order linearity within each segment. While this segmentation increases the minimum number of generation steps to match the number of segments, it substantially reduces both training cost and time. When compared to the previous segment-based rectified flow method, PeRFlow (Yan et al., ), our approach demonstrates significantly better performance in experiments conducted on Stable Diffusion v1-5 (Rombach et al., ) and Stable Diffusion XL (Podell et al., ).

We summarize our main contributions as follows: (i) We conduct an in-depth analysis of the essence of rectification and extend rectified flow to rectified diffusion. (ii) We identify that it is not straightness but first-order property is the essential training target of rectified diffusion with theoretical derivations. (iii) Comprehensive comparisons on rectification, distillation and phased OED segmentation demonstrate our method achieves superior trade-off between generation quality and training efficiency over rectified flow-based models.

Figure: Figure 2: Training iterations. 1-step performance of Rectified Diffusion significantly surpasses the 1-step performance of Rectified Flow within only 20,000 iterations with batch size 128 (8% trained images of Rectified Flow) and consistently grows with more training iterations.
Refer to caption: x2.png

## 2 Rectified Diffusion: Generalizing the design space of rectified flow into general diffusion models

#### Rectified flow is a subset of rectified diffusion.

In the following discussion, we apply the diffusion form $\mathbf{x}_{t}=\alpha_{t}\mathbf{x}_{0}+\sigma_{t}\bm{\epsilon}$ to introduce rectified diffusion. Note that this form of diffusion covers the flow-matching since we can simply set $\alpha_{t}=1-t$ and $\sigma_{t}=t$. Considering the different prediction types, we apply the epsilon-prediction for the following discussion. But note that different prediction types can be converted effortlessly through re-parameterization. For $\mathbf{x}_{0}$-prediction, we have $\mathbf{x}_{0}=\frac{\mathbf{x}_{t}-\sigma_{t}\bm{\epsilon}}{\alpha_{t}}$. For $\bm{v}$-prediction utilized in rectified flow, we have $\bm{v}=\mathbf{x}_{0}-\bm{\epsilon}=\frac{\mathbf{x}_{t}-(\alpha_{t}+\sigma_{t
})\bm{\epsilon}}{\alpha_{t}}$.
Hence, we claim that rectified flow is a subset of rectified diffusion, and rectified diffusion is a generalization of rectified flow.

### 2.1 The nature of rectification is the retraining with pre-computed noise-sample pair

#### The secret of rectification is using paired noise-sample for training.

To illustrate the differences clearly, we visualize the training processes for standard flow matching and rectification (reflow) training, as described in Algorithm 1 and Algorithm 3, respectively. Differences are highlighted in red. A key observation is that in standard flow matching training, $\mathbf{x}_{0}$ represents real data randomly sampled from the training set, while the noise $\bm{\epsilon}$ is also randomly sampled from Gaussian. This results in random pairing between noise and sample. In contrast, in rectification training, the noise is pre-sampled from Gaussian, and the images are generated using pre-sampled noise by the model from the previous round of rectification (the pre-trained model), leading to a deterministic pairing.

#### Flow-matching training is a subset of standard diffusion training.

In addition, Algorithm 2 visualizes the training process of a more general diffusion model, with differences to Algorithm 1 highlighted in blue and orange. It’s important to note that flow matching is a specific case of the diffusion forms we discuss. From the algorithms, it is evident that the only distinctions between them lie in the diffusion form and prediction type. Consequently, flow matching training is just a special case of general diffusion training under a particular diffusion form and prediction type.

By comparing Algorithms 2 and 3 with Algorithm 1, it is straightforward to derive Algorithm 4. Essentially, by incorporating the pre-trained model to collect noise-sample pairs and replacing the randomly sampled noise and real samples with these pre-collected pairs in the general diffusion training, we obtain the training algorithm for rectified diffusion.

Figure: Algorithm 1 Flow Matching $\bm{v}$-Prediction

Figure: Algorithm 2 Diffusion Training $\bm{\epsilon}$-Prediction

Figure: Algorithm 3 Rectified Flow $\bm{v}$-Prediction

Figure: Algorithm 4 Rectified Diffusion $\bm{\epsilon}$-Prediction

### 2.2 Understanding the first-order ODE ( ⋆ ⁣ ⋆ ⁣ ⋆ ⋆ ⋆ ⋆ \star\star\star ⋆ ⋆ ⋆ )

For the above discussed general diffusion form $\mathbf{x}_{t}=\alpha_{t}\mathbf{x}_{0}+\sigma_{t}\bm{\epsilon}$, there exists an exact ODE solution form (Lu et al., ),

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}{\color[rgb]{0,0,1}\bm{ \epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{\lambda}},{t_{\lambda}})}\mathrm{d} \lambda\,,$ (1) $$

where $\lambda_{t}=\ln\frac{\alpha_{t}}{\sigma_{t}}$, and $t_{\lambda}$ is the inverse function of $\lambda_{t}$. The left term $\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}$ is a pre-defined deterministic scaling. The right term is the exponentially weighted integral of epsilon predictions. The first order ODE means the above integral with arbitrary $t$ and $s$ is equivalent to

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}{\color[rgb]{1,0,0}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s)}\int_{ \lambda_{s}}^{\lambda_{t}}e^{-\lambda}\mathrm{d}\lambda=\frac{\alpha_{t}}{ \alpha_{s}}\mathbf{x}_{s}+\alpha_{t}{\color[rgb]{1,0,0}\bm{\epsilon}_{\bm{ \theta}}(\mathbf{x}_{s},s)}(\frac{\sigma_{t}}{\alpha_{t}}-\frac{\sigma_{s}}{ \alpha_{s}})\,.$ (2) $$

We show that the equivalent of Equation 1 and Equation 2 for arbitrary $t$ and $s$ holds and only holds if the epsilon prediction on the same ODE trajectory is a constant in Thereom 1.

#### First-order ODE has the same form of predefined diffusion form.

To put it in a nutshell, we assume the ODE trajectory is a first-order ODE, and there exists a solution point $\mathbf{x}_{0}$. Therefore, the epsilon predictions on the ODE trajectory with solution point $\mathbf{x}_{0}$ are constant, which we denote as $\bm{\epsilon}$. Substitute $s=0$, $\mathbf{x}_{0}$, $\alpha_{s}=1$, $\sigma_{s}=0$ and $\bm{\epsilon}$ into Equation 2, we have

$$ $\displaystyle~{}\mathbf{x}_{t}=\alpha_{t}\mathbf{x}_{0}+\sigma_{t}\bm{\epsilon}.$ (3) $$

This has exactly the same form of predefined forward process. Therefore, we have that the first-order ODE is exactly the weighted interpolation of data and noise following predefined forward diffusion form. The only difference is that, the $\bm{\epsilon}$ and $\mathbf{x}_{0}$ in the above equation is deterministic pair on the same ODE trajectory. While, for standard diffusion training, the $\mathbf{x}_{0}$ and $\bm{\epsilon}$ are randomly sampled.
That indicates that if we achieve perfect coupling of data $\mathbf{x}_{0}$ and noise $\bm{\epsilon}$ at training, and there’s no intersections among different paths (otherwise the epsilon predictions can be the epsilon prediction expectation of different paths), the trained diffusion models in the ideal case (without optimization error) will obtain the first-order ODE.

#### First-order ODE supports consistent generation with arbitrary inference steps.

Additionally, note that if the epsilon predictions on the same trajectory are constant, it is easy to show that the $\mathbf{x}_{0}$-predictions are also constant. Therefore, the first-order ODE can flexibly support one-step generation ($\mathbf{x}_{T}\rightarrow\mathbf{x}_{0}$) or multi-step generation ($\mathbf{x}_{T}\rightarrow\dots\rightarrow\mathbf{x}_{0}$). If a perfect first-order ODE is achieved, we will always get indentical generation results with arbitrary inference steps.

Figure: (a) Flow Matching
Refer to caption: x3.png

#### First-order ODE can be inherently curved.

For the first-order ODE, though the trajectories of flow-matching based methods are straight, the trajectories of other forms of diffusion models can be inherently curved. But if we define $\mathbf{y}_{t}=\frac{\mathbf{x}_{t}}{\sigma_{t}}$, we will have $\mathbf{y}_{t}=\frac{\alpha_{t}}{\sigma_{t}}\mathbf{x}_{0}+\bm{\epsilon}$ from the Equation 3. We can easily obeserve that the trajectory of $\mathbf{y}_{t}$ is a straight line from the initial point $\bm{\epsilon}$ towards the direction of $\mathbf{x}_{0}$ (i.e, first-order trajectories can be converted to straight lines). We showcase our findings in Fig. 3.We select $\mathbf{x}_{0}=[0,1]$ and $\bm{\epsilon}=[1,1]$ The Fig. 3(a) and Fig. 3(b) show the first-order trajectory of flow-matching and EDM. They are both straight, but EDM has a totally different trajectory and magnitude. Fig. 3(c) and Fig. 3(d) show the first-order trajectory of DDPM and Sub-VP. Their first-order trajectory are inherently curved. Fig. 3(e) shows the trajectory of $\mathbf{y}_{t}=\frac{\alpha_{t}}{\sigma_{t}}\mathbf{x}_{0}+\bm{\epsilon}$. It shows that all the first-order trajectories can be converted into straight lines with simple timestep-dependent scaling.

### 2.3 Rectified Diffusion (Phased)

Completely linearizing the ODE path of a pre-trained diffusion model is challenging because the original ODE can deviate significantly from a first-order form. In Fig. 4, we visualize both the original diffusion ODE path and the corresponding first-order ODE path. Since it’s hard to intuitively determine whether a curved ODE path satisfies first-order linearity, we represent the first-order ODE path with a straight line. A significant gap between the two paths is evident. However, enforcing local first-order linearity is more feasible. As shown on the right side of the figure, when the ODE path is divided into two segments along the time axis and each segment is linearized separately, the new ODE path is closer to the original one. This observation motivates the development of our rectification diffusion (phased).

We set intermediate time steps as $s_{0}=0<s_{1}<s_{2}<\cdots<s_{M-1}=t_{\max}$ along the time axis of ODE, where $M$ is the number of phases. The training process begins with sampling $\mathbf{x}_{0}$ from real data, followed by adding random noise at time step $s_{m}$ to obtain $\mathbf{x}_{s_{m}}$. We then use the pre-trained diffusion model to perform multi-step numerical solving to obtain $\mathbf{x}_{s_{m-1}}$ for the previous intermediate step. However, the phasing idea involves two challenges: 1) determining the noise $\bm{\epsilon}$ corresponding to the first-order path, and 2) determine the sample $\mathbf{x}_{t}$ at any time $t$ between $s_{m}$ and $s_{m-1}$ on the same first-order ODE, where $t\in(s_{m-1},s_{m})$.

Fortunately, the transition formula between any two points on the first-order ODE is known, as shown in Equation 2. Through a simple transformation, we have the noise $\bm{\epsilon}$ corresponding to the ODE path between $\mathbf{x}_{s_{m}}$ and $\mathbf{x}_{s_{m-1}}$ can be expressed as:

$$ $\displaystyle\bm{\epsilon}=\frac{\frac{\mathbf{x}_{s_{m-1}}}{\alpha_{s_{m-1}}} -\frac{\mathbf{x}_{s_{m}}}{\alpha_{s_{m}}}}{\frac{\sigma_{s_{m-1}}}{\alpha_{s_ {m-1}}}-\frac{\sigma_{s_{m}}}{\alpha_{s_{m}}}}=\frac{\Delta\mathbf{z}}{\Delta \text{NSR}}\,,$ (4) $$

where $\Delta\mathbf{z}$ represents the change in $\mathbf{z}_{t}=\frac{\mathbf{x}_{t}}{\alpha_{t}}$, and $\Delta\text{NSR}$ denotes the change in $\frac{\sigma_{t}}{\alpha_{t}}$.
Once this noise $\bm{\epsilon}$ is calculated, it can be directly substituted into Equation 2 to compute $\mathbf{x}_{t}$ at any time $t$ along the ODE path.

Figure: Figure 4: ODE trajectory comparison of diffusion models, rectified diffusion models, and phased consistency models. We apply straight lines for more clear demonstration. The solid line shows the original diffusion ODE path, while the dashed line shows the rectified ODE path.
Refer to caption: x8.png

### 2.4 Rectified Diffusion facilitates the consistency distillation

Previous work (Liu et al., ) proposes applying naive distillation after rectification to enhance one-step generation ability. This is because, after rectification, the model cannot achieve a perfect first-order path due to issues like optimization, model capacity, and ODE path intersections. As a result, rectified flow-based methods still do not perform as well as the most advanced distillation methods at low-step regime (e.g., 1-step generation). Following it, we also utilize distillation to further improve the model’s performance at low-step regime after rectified diffusion. Instead of using naive distillation, we employ the more advanced distillation technique–consistency distillation (Song et al., ), which eliminates the need to regenerate large numbers of samples. Moreover, we found that after rectification, where the ODE path is approximately first-order, consistency distillation leads to significantly faster training and better performance. This is because the training objective of a first-order ODE imposes a stronger constraint than self-consistency. In Fig. 5, we illustrate the differences between the diffusion model, consistency model, and rectified diffusion. The consistency model only adjusts the direction of the model’s predictions without altering the ODE path itself, while rectified diffusion enforces a change in the ODE path to a first-order form.

Figure: (a) Diffusion models
Refer to caption: extracted/5920288/figs/cm/diffusion.png

## 3 Empirical Validation

Figure: Figure 6: Effectiveness of Classifier-Free Guidance.
Refer to caption: x9.png

### 3.1 Validation Setup

To thoroughly compare our approach with rectified flow-based methods, we organize the comparison into three levels:

- 1)
Rectification Comparison: InstaFlow (Liu et al., ) proposes initializing a v-prediction-based flow-matching model using Stable Diffusion v1-5 (Rombach et al., ), followed by further training with their rectified flow method, which we refer to as Rectified Flow. To compare with this, we apply the rectified diffusion method to continue training Stable Diffusion v1-5, referred to as Rectified Diffusion. This comparison aims to demonstrate the faster training speed and superior performance of our proposed rectified diffusion approach.
- 2)
Distillation Comparison: In the InstaFlow paper, the authors suggest using a standard distillation technique to improve the model’s performance in a one-step scenario, which we refer to as Rectified Flow (Distill). Similarly, we apply a distillation strategy to enhance performance at low-step regimes, specifically using consistency distillation to boost training efficiency. This approach is termed Rectified Diffusion (CD).
- 3)
Phased ODE Segmentation: PeRFlow (Yan et al., ) introduces the concept of segmenting the ODE and presents experimental results on both SD and SDXL (Podell et al., ), termed PeRFlow and PeRFlow-XL, respectively. We extend this idea by proposing a method for phasing the ODE to enforce first-order property within each sub-phase, which we call Rectified Diffusion (Phased) and Rectified Diffusion-XL (Phased).

Across all three of these comparative experiments, our methods demonstrate significantly superior performance.

**Table 1: Performance comparison on validation set of COCO-2017.**
| Method | Res. | Time ($\downarrow$) | # Steps | # Param. | FID ($\downarrow$) | CLIP ($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- |
| SDv1-5+DPMSolver (Upper-Bound) (Lu et al., ) | 512 | 0.88s | 25 | 0.9B | 20.1 | 0.318 |
| Rectified Flow (Liu et al., ) | 512 | 0.88s | 25 | 0.9B | 21.65 | 0.315 |
| Rectified Flow (Liu et al., ) | 512 | 0.09s | 1 | 0.9B | 47.91 | 0.272 |
| Rectified Flow (Liu et al., ) | 512 | 0.13s | 2 | 0.9B | 31.35 | 0.296 |
| Rectified Diffusion (Ours) | 512 | 0.09s | 25 | 0.9B | 21.28 | 0.316 |
| Rectified Diffusion (Ours) | 512 | 0.09s | 1 | 0.9B | 27.26 | 0.295 |
| Rectified Diffusion (Ours) | 512 | 0.13s | 2 | 0.9B | 22.98 | 0.309 |
| Rectified Flow (Distill) (Liu et al., ) | 512 | 0.09s | 1 | 0.9B | 23.72 | 0.302 |
| Rectified Flow (Distill) (Liu et al., ) | 512 | 0.13s | 2 | 0.9B | 73.49 | 0.261 |
| Rectified Flow (Distill) (Liu et al., ) | 512 | 0.21s | 4 | 0.9B | 103.48 | 0.245 |
| Rectified Diffusion (CD) (Ours) | 512 | 0.09s | 1 | 0.9B | 22.83 | 0.305 |
| Rectified Diffusion (CD) (Ours) | 512 | 0.13s | 2 | 0.9B | 21.66 | 0.312 |
| Rectified Diffusion (CD) (Ours) | 512 | 0.21s | 4 | 0.9B | 21.43 | 0.314 |
| PeRFlow (Yan et al., ) | 512 | 0.21s | 4 | 0.9B | 22.97 | 0.294 |
| Rectified Diffusion (Phased) (Ours) | 512 | 0.21s | 4 | 0.9B | 20.64 | 0.311 |
| PeRFlow-SDXL (Yan et al., ) | 1024 | 0.71s | 4 | 3B | 27.06 | 0.335 |
| Rectified Diffusion-SDXL (Phased) (Ours) | 1024 | 0.71s | 4 | 3B | 25.81 | 0.341 |

Figure: (a) 1-step comparison
Refer to caption: x10.png

**Table 2: Performance comparison on COCO-2014.**
| Method | Res. | Time ($\downarrow$) | # Steps | # Param. | FID ($\downarrow$) | CLIP ($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- |
| Autoregressive Models |  |  |  |  |  |  |
| DALL·E (Ramesh et al., ) | 256 | - | - | 12B | 27.5 | - |
| CogView2 (Ding et al., ) | 256 | - | - | 6B | 24.0 | - |
| Parti-750M (Yu et al., ) | 256 | - | - | 750M | 10.71 | - |
| Parti-3B (Yu et al., ) | 256 | 6.4s | - | 3B | 8.10 | - |
| Parti-20B (Yu et al., ) | 256 | - | - | 20B | 7.23 | - |
| Make-A-Scene (Gafni et al., ) | 256 | 25.0s | - | - | 11.84 | - |
| Masked Models |  |  |  |  |  |  |
| Muse (Chang et al., ) | 256 | 1.3 | 24 | 3B | 7.88 | 0.32 |
| Diffusion Models |  |  |  |  |  |  |
| GLIDE (Nichol et al., ) | 256 | 15.0s | 250 | 5B | 12.24 | - |
| DALL·E 2 (Ramesh et al., ) | 256 | - | 250+27 | 5.5B | 10.39 | - |
| LDM (Rombach et al., ) | 256 | 3.7s | 250 | 1.45B | 12.63 | - |
| Imagen (Saharia et al., ) | 256 | 9.1s | - | 3B | 7.27 | - |
| eDiff-I (Balaji et al., ) | 256 | 32.0s | 25+10 | 9B | 6.95 | - |
| Generative Adversarial Networks (GANs) |  |  |  |  |  |  |
| LAFITE (Zhou et al., ) | 256 | 0.02s | 1 | 75M | 26.94 | - |
| StyleGAN-T (Sauer et al., ) | 512 | 0.10s | 1 | 1B | 13.90 | $\sim$0.293 |
| GigaGAN (Kang et al., ) | 512 | 0.13s | 1 | 1B | 9.09 | - |
| Stable Diffusion (0.9 B) and its accelerated or distilled versions |  |  |  |  |  |  |
| GANs |  |  |  |  |  |  |
| UFOGen (Xu et al., ) | 512 | 0.09s | 1 | 0.9B | 12.78 | - |
| DMD (CFG=3) (Yin et al., ) | 512 | 0.09s | 1 | 0.9B | 11.49 | - |
| DMD (CFG=8) (Yin et al., ) | 512 | 0.09s | 1 | 0.9B | 14.98 | 0.320 |
| SD-Turbo (Sauer et al., ) | 512 | 0.09s | 1 | 0.9B | 16.59 | 0.312 |
| Distillation |  |  |  |  |  |  |
| BOOT (Gu et al., ) | 512 | 0.09s | 1 | 0.9B | 48.20 | 0.26 |
| Guided Distillation (Meng et al., ) | 512 | 0.09s | 1 | 0.9B | 37.3 | 0.27 |
| LCM (Luo et al., ) | 512 | 0.09s | 1 | 0.9B | 37.3 | 0.27 |
| Phased Consistency Model (Wang et al., ) | 512 | 0.09s | 1 | 0.9B | 17.91 | 0.296 |
| Phased Consistency Model (Wang et al., ) | 512 | 0.21s | 4 | 0.9B | 11.70 | - |
| SiD-LSG ($\kappa=4.5$) | 512 | 0.09s | 1 | 0.9B | 16.59 | 0.317 |
| SiD-LSG ($\kappa=3$) | 512 | 0.09s | 1 | 0.9B | 13.21 | 0.314 |
| SiD-LSG ($\kappa=2$) | 512 | 0.09s | 1 | 0.9B | 9.56 | 0.313 |
| SiD-LSG ($\kappa=1.5$) | 512 | 0.09s | 1 | 0.9B | 8.71 | 0.302 |
| SiD-LSG ($\kappa=4.5$) | 512 | 0.09s | 1 | 0.9B | 16.59 | 0.317 |
| Rectification ($\star\star\star$) |  |  |  |  |  |  |
| SDv1-5+DPMSolver (Upper-Bound) (Lu et al., ) | 512 | 0.88s | 25 | 0.9B | 9.78 | 0.318 |
| Rectified Flow (Liu et al., ) | 512 | 0.88s | 25 | 0.9B | 11.34 | 0.313 |
| Rectified Flow (Liu et al., ) | 512 | 0.09s | 1 | 0.9B | 36.68 | 0.272 |
| Rectified Flow (Liu et al., ) | 512 | 0.13s | 2 | 0.9B | 20.01 | 0.296 |
| Rectified Diffusion (Ours) | 512 | 0.88s | 25 | 0.9B | 10.73 | 0.315 |
| Rectified Diffusion (Ours) | 512 | 0.09s | 1 | 0.9B | 16.88 | 0.293 |
| Rectified Diffusion (Ours) | 512 | 0.13s | 2 | 0.9B | 12.57 | 0.307 |
| Rectified Flow (Distill) (Liu et al., ) | 512 | 0.09s | 1 | 0.9B | 13.67 | 0.302 |
| Rectified Flow (Distill) (Liu et al., ) | 512 | 0.13s | 2 | 0.9B | 62.81 | 0.261 |
| Rectified Diffusion (CD) (Ours) | 512 | 0.09s | 1 | 0.9B | 12.54 | 0.303 |
| Rectified Diffusion (CD) (Ours) | 512 | 0.13s | 2 | 0.9B | 11.41 | 0.310 |
| PeRFlow (Yan et al., ) | 512 | 0.09s | 1 | 0.9B | 18.59 | 0.264 |
| Rectified Diffusion (Phased) (Ours) | 512 | 0.09s | 1 | 0.9B | 10.21 | 0.310 |
| Stable Diffusion XL (3B) and its accelerated or distilled versions |  |  |  |  |  |  |
| GANs |  |  |  |  |  |  |
| SDXL-Turbo Sauer et al. () | 512 | 0.15s | 1 | 3B | 24.57 | 0.337 |
| SDXL-Turbo Sauer et al. () | 512 | 0.34s | 4 | 3B | 23.19 | 0.334 |
| SDXL-Lightning (Lin et al., ) | 1024 | 0.35s | 1 | 3B | 23.92 | 0.316 |
| SDXL-Lightning (Lin et al., ) | 1024 | 0.71s | 4 | 3B | 24.56 | 0.323 |
| DMDv2 (Yin et al., ) | 1024 | 0.35s | 1 | 3B | 19.01 | 0.336 |
| DMDv2 (Yin et al., ) | 1024 | 0.71s | 4 | 3B | 19.32 | 0.332 |
| Distillation |  |  |  |  |  |  |
| LCM (Luo et al., ) | 1024 | 0.35s | 1 | 3B | 81.62 | 0.275 |
| LCM (Luo et al., ) | 1024 | 0.71s | 4 | 3B | 22.16 | 0.317 |
| Phased Consistency Model (Wang et al., ) | 1024 | 0.35s | 1 | 3B | 25.31 | 0.318 |
| Phased Consistency Model (Wang et al., ) | 1024 | 0.71s | 4 | 3B | 21.04 | 0.329 |
| Rectification ($\star\star\star$) |  |  |  |  |  |  |
| PeRFlow-XL (Yan et al., ) | 1024 | 0.71s | 4 | 3B | 20.99 | 0.334 |
| Rectified Diffusion-XL (Phased) (Ours) | 1024 | 0.71s | 4 | 3B | 19.71 | 0.340 |

### 3.2 Comparison

Training cost.
Following the setup from the InstaFlow paper, we first use Stable Diffusion v1-5 and DPM-Solver (Lu et al., ) to generate 1.6 million images. Since InstaFlow does not specify the prompts used, we generate images using a randomly sampled set of 1.6 million prompts. During the training of Rectified Diffusion, we used a batch size of 128 for a total of 200,000 iterations, resulting in a total of $128\times 200,000=25,600,000$ samples processed. In comparison, InstaFlow processed $64\times 70,000+1024\times 25,000=30,080,000$ samples. Thus, our total training cost is lower than that of InstaFlow. Additionally, InstaFlow’s total training time was 75.2 A100 GPU days, whereas our method required approximately 20 A800 GPU days. Typically, the training efficiency of an A800 is about 80% of that of an A100. We attribute this significant reduction in training time to not using the LPIPS Loss (Zhang et al., ), which generally improves FID but incurs substantial memory and computational costs during the latent diffusion decoding process.
For the second-stage distillation, we employ consistency distillation training with a batch size of 512 for 10,000 iterations, consuming a total of 4.6 A800 GPU days. In contrast, the distillation process described in the InstaFlow paper takes 110 A100 GPU days. Our training cost is approximately 3% of the GPU days of InstaFlow’s distillation process.

Training speed.
We monitor the performance of Rectified Diffusion in terms of FID and CLIP score at different stages of training. It was observed from Fig. 2 that our method achieve superior one-step performance compared to Rectified Flow after just 20,000 iterations, with further significant improvements as training continued. At this stage, the number of samples processed was only about 8% of the samples processed by Rectified Flow. This efficiency is largely because Rectified Diffusion does not require converting the original epsilon prediction diffusion model, which follows the DDPM form, into a v-prediction flow-matching model—a process that incurs significant computational cost.

Qualitative comparison.
We present a comparison of the images generated by Rectified Diffusion and Rectified Flow across various scenarios in Fig. 8 and Fig. 9. First, we can observe that the Rectified Flow model performs poorly at low step counts, producing only very blurry images in fewer than eight steps. Additionally, we notice that the images generated by PeRFlow are blurry and fail to reflect the content of the text. Moreover, the results generated by Rectified Flow (Distill) remain relatively blurry and lack the ability for multi-step refinement, which limits its applicability. Rectified Diffusion shows clearly superiority in these settings.

Quantitative comparison. We calculate the FID (Heusel et al., ) and CLIP scores (Radford et al., ) for different models on the COCO-2017 validation set (Lin et al., ) and the 30k subset of the COCO-2014 validation set (Lin et al., ), respectively. As shown in Table 1 and Table 3, our model consistently outperforms the methods based on rectified flow across both metrics, different scenarios, and various steps. It also achieves performance comparable to advanced distillation and GAN training methods.

Human preference metrics. To more comprehensively evaluate the model performance, we compare the outputs using human preference models. We follow the testing setup of Diffusion-DPO (Wallace et al., ), generating images with 500 unique prompts from the Pick-a-pic (Kirstain et al., ) validation set for comparison. We used the Laion-Aesthetic Predictor (Schuhmann, ), Pickscore (Kirstain et al., ), HPSv2 (Wu et al., ), and ImageReward (Xu et al., ) to score the generated results from each model individually and calculate the win rate of each model across these metrics. Our results, shown in Fig 7, consistently outperform the results of Rectified Flow-based models.

CFG-influence.
We show the performance comparison of FID and CLIP Score between Rectified Flow and Rectified Diffusion under different step counts and CFG values in Fig. 6. We observe that Rectified Diffusion consistently outperforms Rectified Flow, especially in the low-step regime. Additionally, we find that CFG has a significant impact on both Rectified Diffusion and Rectified Flow; even in the 1-step generation scenario, using an appropriate CFG value can still significantly enhance performance.

## 4 Conclusion

In conclusion, we rethink and investigate the essence of rectified flow. We demonstrate that retraining with pre-collected noise-image pairs is the most important factor. Building on this insight, we propose Rectified Diffusion, extending its scope to general diffusion forms. We identify that it is not straightness but first-order property is the essential training target of Rectified Diffusion. Additionally, by incorporating consistency distillation and introducing Rectified Diffusion (Phased), we further enhance training efficiency and model performance, offering a streamlined approach to efficient high-fidelity visual generation. Vast validation demonstrates the advancements of Rectified Diffusion.

## Appendix

[appendices]
\printcontents[appendices]l1

## Appendix I Related Works

Diffusion models.
Diffusion models have steadily become the foundational models in image synthesis (Ho et al., ; Song et al., ; Karras et al., ). Extensive research has been conducted to explore their underlying principles (Lipman et al., ; Chen & Lipman, ; Song et al., ; Kingma et al., ) and to expand or enhance the design space of these models (Song et al., ; Karras et al., ; Kingma et al., ). Additionally, several works have focused on innovating the model architecture (Dhariwal & Nichol, ; Peebles & Xie, ), while others have scaled up diffusion models for text-conditioned image synthesis and various real-world applications (Shi et al., ; Rombach et al., ; Podell et al., ). Moreover, efforts to accelerate sampling have been pursued at both the scheduler level (Karras et al., ; Lu et al., ; Song et al., ) and the training level (Meng et al., ; Song et al., ; Zhou et al., ; ). The former typically involves refining the approximation of the PF-ODE (Lu et al., ; Song et al., ), while the latter focuses on distillation techniques (Meng et al., ; Salimans & Ho, ; Song et al., ; Wang et al., ; ) or initializing diffusion weights for GAN training (Sauer et al., ; Lin et al., ; Xu et al., ).

Rectified Flow. Lipman et al. () proposes the flow matching based on continuous normalizing flows, providing a different and unified perspective to understand diffusion models. Liu et al. () proposes the method rectified flow, setting up important baseline for diffusion acceleration and providing a solid theoretical analysis. It proposes rectification to straighten the ODE path of flow-matching based diffusion models. In the proof, Liu et al. () show that the rectification allows for non-decreasing straightness of ODE. Liu et al. () scale up the idea of rectified flow into large text-to-image generations, achieving one-step generation without introducing GAN. Yan et al. () proposes to split the ODE path into multi-phase following the InstaFlow (Liu et al., ). Lee et al. () analysises that one-time rectification is generally enough to achieve pure straightness and proposes better optimization strategy for enhanced performance of rectified flows.

## Appendix II Limitations

At low-step regime, the performance of methods based on rectification still lags behind state-of-the-art methods based on distillation (Zhou et al., ) or GAN training (Yin et al., ; Sauer et al., ). Additional distillation steps are needed to improve low-step performance, which is also stated in InstaFlow (Liu et al., ).

## Appendix III Proof for first-order ODE

###### Theorem 1

For the general diffusion form $\mathbf{x}_{t}=\alpha_{t}\mathbf{x}_{0}+\sigma_{t}\bm{\epsilon}$, there exists an exact ODE solution form as follows:

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\bm{\epsilon}_{\bm{\theta}}( \mathbf{x}_{t_{\lambda}},t_{\lambda})\mathrm{d}\lambda,$ (5) $$

where $\lambda_{t}=\ln\frac{\alpha_{t}}{\sigma_{t}}$ and $t_{\lambda}$ is the inverse function of $\lambda_{t}$. The first-order ODE satisfies

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s)\int_{\lambda_{s}}^{\lambda_{ t}}e^{-\lambda}\mathrm{d}\lambda=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}- \alpha_{t}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s)(\frac{\alpha_{s}}{ \sigma_{s}}-\frac{\alpha_{t}}{\sigma_{t}})\,.$ (6) $$

We show the equivalence between Equation 5 and Equation 6 for arbitrary $t$ and $s$, which holds true if and only if $\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t},t)$ is constant.

###### Proof 1

If $\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t},t)$ is constant, then the Equation 5 and Equation 6 are equivalent.

Assumption: Let $\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s)=\bm{\epsilon}_{0}$ be a constant.

Substituting $\bm{\epsilon}_{0}$ into the Equation 5:

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}\bm{\epsilon}_{0}\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\mathrm{d}\lambda$ (7) $$

Calculating the integral:

$$ $\displaystyle\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\mathrm{d}\lambda=e^{ -\lambda_{s}}-e^{-\lambda_{t}}=\frac{\sigma_{s}}{\alpha_{s}}-\frac{\sigma_{t}} {\alpha_{t}}$ (8) $$

Substituting the result:

$$ $\displaystyle\mathbf{x}_{t}=\frac{\alpha_{t}}{\alpha_{s}}\mathbf{x}_{s}-\alpha _{t}\bm{\epsilon}_{0}\left(\frac{\sigma_{s}}{\alpha_{s}}-\frac{\sigma_{t}}{ \alpha_{t}}\right)$ (9) $$

Comparing with the equation: The results match, thus proving equivalence.

If Equation 5 and Equation 6 are equivalent, then $\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t},t)$ must be constant.

Assumption: Assume the two are equivalent:

$$ $\displaystyle-\alpha_{t}\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\bm{ \epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{\lambda}},t_{\lambda})\mathrm{d}\lambda =-\alpha_{t}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s)\int_{\lambda_{s}}^{ \lambda_{t}}e^{-\lambda}\mathrm{d}\lambda$ (10) $$

Removing the constant factor:

$$ $\displaystyle\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\bm{\epsilon}_{\bm{ \theta}}(\mathbf{x}_{t_{\lambda}},t_{\lambda})\mathrm{d}\lambda=\bm{\epsilon}_ {\bm{\theta}}(\mathbf{x}_{s},s)\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda} \mathrm{d}\lambda$ (11) $$

Differentiating with respect to $t$ with Newton-Leibniz theorem:

$$ $\displaystyle\frac{d}{dt}\left(\int_{\lambda_{s}}^{\lambda_{t}}e^{-\lambda}\bm {\epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{\lambda}},t_{\lambda})\mathrm{d} \lambda\right)=e^{-\lambda_{t}}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{ \lambda}},t_{\lambda})\frac{d\lambda_{t}}{dt}$ (12) $$

Comparing both sides:

$$ $\displaystyle e^{-\lambda_{t}}\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{ \lambda}},t_{\lambda})\frac{d\lambda_{t}}{dt}=\bm{\epsilon}_{\bm{\theta}}( \mathbf{x}_{s},s)e^{-\lambda_{t}}\frac{d\lambda_{t}}{dt}$ (13) $$

Since $\frac{d\lambda_{t}}{dt}\neq 0$ and $e^{-\lambda_{t}}>0$, we can cancel terms, leading to:

$$ $\displaystyle\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t_{\lambda}},t_{\lambda}) =\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{s},s),\forall t_{\lambda}\in[s,t].$ (14) $$

Conclusion: This shows that for any $t$, $\bm{\epsilon}_{\bm{\theta}}(\mathbf{x}_{t},t)$ must be constant, proving the “if and only if” statement.

###### Theorem 1

###### Proof 1

## Appendix IV More results.

Figure: Figure 8: Qualitative comparison.
Refer to caption: x15.png

Figure: Figure 9: Qualitative comparison.
Refer to caption: x16.png

Figure: Figure 10: Qualitative comparison.
Refer to caption: x17.png

## Peer Reviews

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. The performance metrics on FID shows only marginal improvements over Rectified Flow. The FID metric for Rectified Flow shown in Table 2 corresponding to step2 (62.81) is higher than step1 (13.67). Are these values correct? Generally step2 value should be lower.
2. Could the authors validated the performance of their method on more datasets?
soundness: 3
strengths: 1. The authors  generalized rectified flow to a wider array of diffusion models by focusing on paired noise-sample training, making the approach compatible with different prediction types and diffusion models.
2. The paper demonstrates that Rectified Diffusion performs better than prior methods like InstaFlow and Rectified Flow in terms of FID and CLIP scores, particularly when generating images in low-step regimes.
3. By dividing the ODE path into phases with enforced linearity in each phase, the model maintains a balance between quality and computational efficiency without altering the fundamental diffusion trajectory.
summary: This paper proposes Rectified Diffusion that generalizes rectified flow to broader categories of diffusion models, improving image generation efficiency by removing the need for flow-matching and v-prediction. Key components of the proposed method include pre-computed noise-sample pairs, consistency distillation, and phased ODE segmentation, allowing the model to achieve comparable or superior quality with fewer training steps.
weaknesses: 1.  While the approach seems efficient, the paper builds on existing methods, particularly using paired noise-sample training, phased ODE segmentation and consistency distillation. The lack of entirely new techniques could limit its novelty in diffusion model research.
2. While this method employs distillation techniques that improve inference speed, achieving single-step image generation without pre-trained models could introduce a novel advancement.
3. The performance metrics on FID shows only marginal improvements over Rectified Flow. The FID metric for Rectified Flow shown in Table 2 corresponding to step2 (62.81) is higher than step1 (13.67). Are these values correct? Generally step2 value should be lower.
4. Although the method is compared against multiple models, all experiments are limited to a single dataset (COCO). Additional experiments on other benchmark datasets, such as ImageNet, would help validate the generalizability of the results.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: From eq 13 to eq 14, it asserts that d\lambda_t/dt is not zero, what is that? Or as \lambda_t = ln \alpha_t / \sigma_t, is it always true that this partial derivative is non-zero? 
Table 1, second block, it seems for example Rectified Flow's FID had great improvement from 1 step to 2 steps then to 25 steps, the Rectified Diffusion's FID did not show a great improvement from 1 step to 2 steps then to 25 steps, while the Rectified Diffusion's FID at 25 steps is slightly better than the FID of Rectified Flow at 25 steps. Can authors elaborate on why Rectified Diffusion does not experience a much-improved performance from 1 step to 25 steps? 
Figure 2 is somehow difficult to understand. Why did the FID and CLIP score of Rectified Flow not change with iterations?
Figure 6, the two rightmost plots, why did the curves of Rectified Diffusion go up at 4 and 8 steps? Does it mean the method need to stop at some optimal points before its performance decreases? And if so, how does one determine the proper steps to stop? 
In Figure 7, it seems the biggest difference in preference given by human evaluation is the Phased 4-step results. Does it imply that Phased implementation, which features multiple straight trajectories rather than one single straight trajectory generally, generally gives better results? Also, in Phased implementation, how many phases were used? It seems this information is not given in the paper.
soundness: 3
strengths: First strength is that the paper shows that straightness is necessary as required in rectified flow. It shows that rectified flow is a special case of rectified diffusion, which is what the paper proposed as a better generative model. 
Second strength is the better quantitative results and favorable comparison with other methods.
summary: This paper presents two findings, one is that straightness as required in rectified flow is not necessary, and second is that it proposed Rectified Diffusion, which generalizes the design space and application scope of rectification to diffusion models.
weaknesses: Maybe I am missing something, it seems the paper argues that straightness of the ODE trajectory is not necessary, but its results appear to show a straight path, such as Figure 4 and Figure 5. 
Other weaknesses include that from visual evaluation, it seems that though the rectified diffusion gave better images than rectified flow in most cases, the improvement is not always clear. For example, in Figure 10, the result of rectified diffusion on the sneakers seem that the string missed a buttonhole. Also, in the pajama example, figure patterns on the pajama in the rectified diffusion result are not as sharp as those on the result of rectified flow. In Figure 10, the huskie images generated by both ReFlow and Rec Diffusion seem to have an extra leg, though it is more evident in the image generated by ReFlow.

---

confidence: 4
contribution: 1
initial_rating: 1
presentation: 2
questions: 1. Please correct the typo "obeserve" in Page 5 line 265 and if there is any more.
 2. In Page 5 around line 265, the authors stated that the trajectory of y_t is a straight line from the initial point ϵ towards the direction of x0 (as shown in Figure 3 (e). This is not clearly explained. Also, It seems like a very special case  when they only used X0 = [0, 1]. It would make more sense if the authors generate the trajectories over time for each elements of X_0 to \epsilon.
3. In Page 4, line 205, the authors mentioned that there exists an exact ODE solution form from Lu et al., 2022, but they omitted the solution was  for  what first order ODE. That will easily confuse the readers.
4. Eqn (2)  in Page 4 should be just an linear approximation of curves. You can use a straight line to approximate the curve, but then you can say this linear line is the result of a straight flow ode.
5. Some descriptions seem like claims that need to be further elaborated, for example those in line 250-line 261.
soundness: 1
strengths: They tried to highlight that straightness is not an essential training target for rectification.
summary: This paper proposed to study the essence of rectified flow and proposed Rectified Diffusion.
weaknesses: It was not stated clearly why it is necessary to study whether straightness is not an essential training target for rectification. To me, it was the straightness used in the ode of the rectified flow algorithm that causes the usage of rectified flow, thus it seems trivial that there is no need to use the rectified flow in a ode that does not follow a straight line. 

The diffusion that follows the curve form that the authors proposed is mentioned in some papers including the original rectified flow paper with the formula listed as well. It can return back to DDPM or DDIM; therefore, to me, it seems like recirculating the algorithm.  The authors might want to look into the following references mentioned in the original rectified flow paper in Lu et al., 2022 (Lipman et al., 2022; Albergo & Vanden-Eijnden, 2022; Heitz et al., 2023).Neklyudov et al. (2022).
While the authors meant to straighten the paths that originally were curves, they can be viewed as approximately meanwhile they can also be viewed as the original straight flows.
Although one can choose any ode path, the paper did not prove (or it is not clear) that their choice gains advantage over existing algorithms. As a matter of fact, the authors used 200000 runs to train the algorithm as shown in figure 2. 

Also, Theorem 1 proved in Appendix seems like a similar version of one in (Lu et al., 2022), there is no need to repeat.

As a result of the weaknesses stated here, the title of this paper seems like questionable itself ( seems like a logic flaw?).

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: * In lines 357-358, the authors claim that the first-order ODE objective is a stronger constraint than self-consistency. If it’s the case that first-order ODE => self-consistency, then why would consistency distillation be necessary after rectified flow?
soundness: 3
strengths: * The quantitative and qualitative results (Figs. 8-10) are quite convincing. The authors provide the essential baselines of rectified flow and PeRFlow, as well as additional Stable Diffusion XL-based baselines in Table 2.
* The idea is original yet simple (a strength not a weakness). It is well-backed by a brief theoretical discussion in Section 2.2.
* Besides a few grammatical errors and too-short figure captions, the presentation is good. The text is easy to follow, and the figures are intuitive.
summary: This paper proposes rectified diffusion, a generalization of rectified flow to first-order ODEs. During training, rectified diffusion aims to correct the learned ODE of the pretrained diffusion model to be closer to a first-order ODE, using precomputed noise-data pairs. Thus the authors argue that using paired noise-data examples is the essential component of both rectified flow and rectified diffusion. The authors further propose phased rectified diffusion and consistency distillation on top of rectified diffusion. In their experiments, they demonstrate superior quantitative and qualitative performance over rectified flow and a previous phased ODE segmentation method (PeRFlow).
weaknesses: * It would have been nice to compare to Consistency Models (Song et al. 2023) for one-step generation.
* In Table 1, is there a typo for 25-step Rectified Diffusion (Ours)? It says the time is only 0.09s, which is the same as 1-step Rectified Diffusion (Ours).

Nitpick:
* Figure 4 should have axis labels. 
* Fonts too small to read in Figure 7.
* Figure 6 caption too short and could use more description.

Meta-review: This submission aims to generalise rectification to a wider class of diffusion models via first-order ODEs. Authors propose that precomputed noise-data pairs constitute the essential component (c.f. flow-matching, v-prediction), to enable a learned diffusion ODE to be closer to a first-order ODE. Experimental work can evidence improved image generation efficiency; comparable or superior quality with fewer training steps (c.f. rectified flow, phased ODE segmentation).

AC notes that there are interesting ideas in the paper. The writing quality could be improved, e.g. the way theorems are stated. Reviews allude to questions regarding results supporting claims. A further important point to note is that non-straight interpolating curves are not new. Previous work from the last two years has investigated other approaches to straightening integration paths [1,2]. These works are not mentioned or compared, however they were shown to improve flow-matching models and report experiments with non-linear paths. Both are well cited and so may not be considered obscure.

This paper was discussed at length with the SAC. 

After reviewing the paper, rebuttal and resulting discussions we find that the overall strengths outweigh the weaknesses and recommends acceptance. For the camera-ready version, the authors should incorporate all key results presented in the rebuttal. In particular, the review process highlighted the need to improve clarity and understanding surrounding straight, curved ODE-trajectories, first-order linearity and acknowledgement of key missing works relating to the straightening of integration paths. 

1. Tong et al. Improving and generalizing flow-based generative models with minibatch optimal transport
2. Albergo et al. Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

Decision: Accept (Poster)