Balanced conic rectified flow

Thank you for taking an interest in our work and for your detailed questions.

$**[W1,2] Performance and Comparison**$
Our study focuses on experimentally addressing the instability of the reflow process and proposing a method to mitigate it. As such, we primarily compared with the original rectified flow, as it is more relevant and better aligned with the objectives of our work, rather than other diffusion-based generative models.

$**[W 3] Metrics**$
Thank you for the interesting suggestion. Calculating recall and precision can show how well our method balances diversity and fidelity compared to the baseline and evaluate the use of real pairs in reflow. We will include these metrics in the Appendix by November 25.

$**[W 4] Typos**$
Thank you for pointing this out. We have corrected these issues.
$\bullet$ In line 746-747 ?? => Fig A12 ,... , A25.
$\bullet$ From line 753 to line 768 => Aligned with other pages.

$**[W 5] Theoretical Analysis**$
Thank you for your valuable comment. Our approach focused on addressing numerical errors in the reflow process through experimental observations, making theoretical analysis beyond our scope. As noted in the Limitations, this is a valuable direction for future research.

We hope these explanations resolve the concerns. Further questions or suggestions would be also appreciated.

Thank you once again for your thoughtful feedback and suggestions. We apologize for not being able to provide the additional materials promptly.

$**[1]**$
Apologies for misunderstanding the intention of your first question. We agree with your suggestion that clarifying the position of 1-rectified flow within the context of flow matching-based generative models would indeed be helpful for readers.
Unfortunately, due to space constraints in the main manuscript, we could not include additional diffusion-based generative models in Table 1.
Instead, we have addressed this limitation by reporting the performance of as many diffusion-based generative models as possible in Appendix J, including your suggested models.
In particular, we have included the following representative models:

• Diffusion-based generative models: EDM [1], LSGM [2], PFGM [3]
• Distillation models: PD [4], DFNO [5], SID [6]
• Consistency models: CD [7], CT [7], ICT [8], CTM [9], CTM+GAN [9]

Additionally, we have clarified this point in Table 1 and Section 4.1 (L357) as follows:

Further results clarify the upper bounds of reflow. We provide the CIFAR-10 unconditional generation qualities of pre-trained diffusion models in Appendix J.

Thank you again for your valuable suggestions.

$**[2,3]**$
Thank you again for your thoughtful comments. The following additional materials have been included:

1. $**Slerp Patterns and Lerp (Appendix B)**$
   To provide further experimental insights into the use of Slerp and the noise patterns that increase and then decrease, we conducted additional experiments with the following settings:

• Strictly increasing noise (SI) : Noise increases from 0.006 to 0.13.
• Strictly decreasing noise (SD) : Noise decreases from 0.13 to 0.006.
• Using Lerp : Linear interpolation.

The strictly increasing pattern allows us to analyze generation quality as noise progressively grows, while the strictly decreasing pattern helps us evaluate performance as noise is gradually removed.

FID : $**6.63**$ (Ours) / 6.64 (SI) / 6.70 (SD) / 7.50 (Lerp)
IS : $**8.72**$ (Ours) / 8.48 (SI) / 8.45 (SD) / 8.46 (Lerp)

• Our scheduling achieves higher FID and IS compared to other settings.
• Using Slerp instead of Lerp generally results in higher FID and IS.

2. $**Recall and Precision (Appendix C)**$
   Recall and precision for 2- and 3-rectified flow have been added, showing significant improvement in recall for 1-step sampling compared to the baseline.

$**Multi step sampling(NFE=104)**$

• 2-rectified flow
  Precision : 0.696 (+0.005) / 0.691(ours)
  recall :0.600 / 0.605 (+0.005) (ours)
• 3-rectified flow
  Precision : 0.698 (+0.007) / 0.691(ours)
  recall : 0.592 / 0.599 (+0.007)(ours)

$**1-step sampling**$

• 2-rectified flow
  Precision : 0.695 / 0.687 (+0.008) (ours)
  recall :0.528 / 0.583 ($**+0.055**$) (ours)
• 3-rectified flow
  Precision : 0.682 (+ 0.009) / 0.691 (ours)
  recall : 0.562 / 0.592 ($**+0.03**$) (ours)

It indicate that while the precision of our method is nearly identical to the baseline, its higher recall demonstrates superior coverage of the real data distribution.

3. $** Extreme Number of Reflow Steps (k = 4) (Appendix D)**$

• Our method consistently outperforms the baseline across general reflow steps, not just for 2 and 3.
• 4-Rectified flow, NFE 100
  IS 8.95 / 9.01 (Ours) , FID 4.49 / 4.19 (Ours)
• 4-Rectified flow, NFE 1
  IS 8.59 / 8.80 (Ours), FID 6.58 / 5.66 (Ours)

For more details, please refer to the revised PDF, Appendix B, C, and D.
We hope the additional materials submitted have helped experimentally support the effectiveness of our method. Any further questions or concerns would be gratefully addressed. Thank you.

[1] Karras et el., Eucidating the design space of diffusion-based generative models, 2022.
[2] Vahdat et el., Score-based generative modeling in latent space, 2021.
[3] Xu et el., oisson flow generative models, 2022.
[4] Salimans & Ho, Progressive distillation for fast sampling of diffusion models, 2022.
[5] Zheng et el., Fast sampling of diffusion models via operator learning, 2023.
[6] Zhou et el., Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation,2024.
[7] Song et el., Consistency models, 2023.
[8] Song & Dhariwal., Improved techniques for training consistency models, 2023.
[9] Kim et el., Consistency trajectory models: Learning probability flow ode trajectory of diffusion, 2023.

I have reviewed the authors’ rebuttal, and most of my concerns have been addressed.

I appreciate the efforts the authors put into the rebuttal. However, considering the contribution level—such as the lack of theoretical analysis, the experimental scale, and the impact on future research—I am unable to raise my score to 8 and will maintain it at its current level for weak acceptance.

We appreciate your thoughtful and detailed feedback on our work. We hope these explanations resolve the concerns, and we would greatly appreciate any further questions or suggestions.

$**[W1] property of$v^{-1**(X_1)}
Thank you for raising your concern.
Using real images does not contradict the original purpose because the noise $v^{-1}(X_1)$ generated using $v^{-1}$ satisfies the properties of the rectified flow as well.

Moreover, the original paper also mentions that $v^{-1}$ (or $-v$) can be used optionally, supporting the validity of this approach. To clarify this point further, we have added explanations about $v^{-1}$ and $v^{-1}(X_1)$ in lines 191–193. Thank you for raising this insightful point.

$**[W2] The reason for Slerp**$
Interpolation in noise space using Slerp is a common practice in generative models ([1], [2]).
We agree that clarifying the use of Slerp and comparing it with other methods adds value. We have added an explanation with citations in Lines 233–235 and will include Lerp results in Appendix C by November 25.
Thank you for your feedback.

$**[W3] About Slerp scheduling pattern**$

As noted in the Limitations section, the noise scheduling used with Slerp is heuristic and based on empirical experimentation, not a theoretical foundation. While effective, it may not be optimal.

We agree that analyzing various noise scheduling patterns (e.g., monotonically decreasing, increasing) would provide valuable insights and will include additional results in the appendix by November 25. Thank you for your suggestion.

$**[W4] The impact of ours when$K > 3$**$
Thank you for the suggestion. While rectified flow methods typically use $k=2$ or $k=3$, exploring $k=4$ or higher could evaluate our method’s robustness against model collapse.

Running $k=5$ experiments is challenging within the discussion period because $k=4$ training must be completed first to generate new pairs. We will include $k=4$ results in the appendix B by November 25.

Thank you again for the constructive comments. We will report the results in the appendix once the additional experiments are completed. Your feedback has enriched our work.

[1] Wang & Golland., 2023, Interpolating between images with diffusion models.
[2] Jang et al., 2024, Spherical linear interpolation and text-anchoring for zero-shot composed image retrieval.

Thank you for your constructive feedback. We have carefully reviewed your comments and have incorporated them into our work.

$**[W1] Comments of$X_0$and$X_1$**$
Thank you for your thoughtful suggestion. We have added L093 (marked in red) to clarify $X_0$ and $X_1$ as below.

In image generation tasks, $X_0\sim\pi_0$ and $X_1\sim\pi_1$ are latent noises and real images, sampled from Gaussian distribution and data distribution, respectively.

$**[W2] Issues with figures**$
We apologize for the inconvenience and thank you for finding them out. We updated them as follows.

$\bullet$ The overlap between Figures 2 and 3 is resolved.
$\bullet$ The axes and legends are written with larger fonts in Figures 7, 8 ,9, 10(a), 10(b), 11 and 12.

$**[W3] Quantitative comparison with [1] \**$
Thank you for the insightful comment.
We primarily compared with the original rectified flow rather than other diffusion-based generative models because our focus is issues due to fake pairs in the reflow process.
Our focus was on addressing issues with using only fake pairs in the reflow process, so we primarily compared with the original rectified flow rather than other diffusion-based generative models.
Similarly, we excluded [1] because its key points are orthogonal to ours:
$\bullet$ using EDM weight for pairing fake pairs
$\bullet$ using Huber loss for reflow
$\bullet$ Developing the sampling distribution for linear interpolation between $X_0$ and $X_1$.

Although we agree that comparison with [1] would enrich our paper, it is difficult to compare as follows. We cannot directly compare the measures in [1] because its training configurations differ
(batch size: ours : 256 / Lee : 512; training iterations: ours : 600K / Lee : 800K) from ours.

We cannot train ours on their configuration or theirs in our configuration because it takes more than two weeks due to:

$\bullet$ [1] uses a diffusion-based model with EDM-initialized weights instead of a 1-rectified flow model for pairing.
$\bullet$ Generating millions of pairs with [1]'s configuration.
$\bullet$ Training for several proposed losses applying our methods.

We have added the keypoints of [1] and potential combination in Line 535~537. Which is infeasible within the discussion period.
Thank you for the suggestion.

$**[W4] Experiments on other datasets**$
Unfortunately, the official repo only provides 1-rectified flow checkpoints, and training a comparable 2-rectified flow requires 4M fake pairs, which is infeasible during the rebuttal period.

Instead, we trained on LSUN with the original 2-rectified flow to provide both qualitative and quantitative results including recon and p-recon errors, IVD, and curvature, as shown in Appendix C. We plan to train additional checkpoints from scratch and share them in the camera ready.

$**[W5] Table2**$
Fine-tuned version has lower curvature and IVD than the original, indicating that our method is more straight than the original. And also fine-tuned version has lower recon and p-recon differences between real and fake images, indicating that our method reduces the bias toward fake samples.

To visually show the performance gaps, we have replaced Table 2 with Figure 11 and clarified this in the revised PDF (Lines 449–451). Thank you.

$**[W6]** v^{-1}$ and $-v$
Thank you for your remark. We used $v^{-1}$ to concisely represent the reverse noise for a real pair, allowing for a more general and compact notation. Since $v^{-1} = -v$ in our case, we have added a footnote (Line159) in the revised PDF. We believe this addition improves the paper’s clarity.

$**[W7]**$"curvature" in L410.
Thank you for pointing this out. We have corrected it.

$**[Q1] NFE (110 vs 104)**$
The NFE difference comes from RK sampling adaptively adjusting based on a threshold. We averaged NFE over 50k samples but did not explicitly mention this, which may have caused confusion.
$\bullet$ Add details about the Runge-Kutta method in the revised paper (Lines 323, 344) and citation for tolerances and parameter settings.

Thank you for your feedback.

$**[Q2] Using only real pairs**$
Yes, we have tested. Similar results to those observed in Appendix B show that this improves 1-step generation but not many-step generation, likely because real images alone cannot fully rewire the velocity field of 1-rectified flow.

This raises important questions about the optimal real-to-fake pair ratio and the feasibility of straightening the velocity field with only real pairs, which are promising directions for future research. Thank you for your insightful question.

Thank you for providing a detailed response and good suggestions. Your feedback has been helpful in clarifying the content of the paper and addressing problematic figures. We hope that the issues you pointed out have been addressed and would greatly appreciate any further thoughts or suggestions you may have.

Thank you for your detailed and thoughtful feedback. We have thoroughly considered your suggestions and integrated them into our work. As there is a lot to address, I will provide my responses in two parts.

$**[W1,2]**$
Thank you for raising your concern.
Fortunately, it is not too strong as follows. $v_1(X_0)$ drifts away from $X_1$. $v_2(X_0)$ drifts away from $v_1(X_0)$ where $v_2$ is 2-rectified flow. Among the possible choices of $v_2(X_0)$, the possibility of being closer to $X_1$ than $v_1(X_0)$ is very small.
Likewise, discrepancies between the reconstruction errors of real and fake images is obvious because the fake samples are not equal to the real samples. The optimal solution of $v_2$ with the fake samples as supervision is $v_2(X_0)=v_1(X_0)$, not $v_2(X_0) = X_1$.

$**[W3]**$
Thank you for raising your curiosity.

• Reconstruction error measures the drift from the original samples.
• The 2-rectified flow makes higher reconstruction error on the real samples than the fake samples.
• Hence, it indicates the 2-rectified flow drifts away from the real samples.

We suppose that the order of the comparative brought confusion on this simple syllogism. We have fixed the sentence to be:

$L_2^\text{recon}$ is higher at the real samples than the fake samples.(L144)

$**[W4]**$
Thank you for raising your curiosity.
Figures 2 and 3(b) suggest that the noise space for fake samples is well-structured, but the manifold for real samples remains unstable, making them more sensitive to perturbations. As a result, the model captures fake sample structure effectively but struggles with real samples.

Thank you for raising your concern

$**[W5]**$
As correctly noted, the original rectified flow provides the option to use reverse trajectories for pairing, and our method builds on this. However, while the original rectified flow just mentions that reverse trajectories can be inferred using $-v$, Our approach goes further by incorporating both real and fake pairs into the reflow process and applying Slerp to $v^{-1}(X_1)$.

The advantages of Slerp are as follows:

• Slerp helps maintain the trajectory and neighborhood between reverse noise corresponding to real images more stably. This reduces $L_2^{recon}$ and $L_2^{p-recon}$ errors for real images.

• Better generation quality(FID, IS) than no-Slerp settings.

To clarify, we have added a note in Line 192 of the revised paper acknowledging the original paper's introduction of this option.

$**[W6]**$
Thank you for raising your concern.

• The perturbed reconstruction error for real samples measures the reconstruction error of perturbed versions of real samples.
• Our training method perturbs the reversed noise from real images, not the images.

Hence, they do not contradict each other.

$**[W7]**$
Thank you for the insightful observation.
This issue stems from a fundamental limitation shared by rectified flow methods that rely on fake images during the reflow process.

• We have added this limitation in Lines 526–530 of the revised paper.

Nevertheless, it is important to note that our method improves the original reflow process, which is the core contribution of this work.
Thank you for your valuable feedback.

$**[W8,9]**$
Thank you for raising your concern.
We use "fake pair" and "generated pair" interchangeably, while "real pair" refers specifically to real images $X_1$​ and $v^{-1}(X_1)$. However, we acknowledge that using terms like "generated image" without clarification after defining "fake pair" might cause confusion for readers.
We have made the following revisions:

• Replaced "generated image" with "fake image" (Line 416, Figure 3(b) + caption, Line 415).
• Added further clarification about "fake," "real," and "generated" in Lines 197–199.

Thank you for the insightful suggestion.

$**[W10]**$
We apologize for the confusion.
The referenced part should have pointed to Figure 9(b) rather than Figure 3(b). This was an error.

As noted in Lines 151–154 of the revised version, the current rectified flow shows lower reconstruction errors for fake pairs than real pairs, indicating a bias toward fake pairs in the original reflow methods. Our claim is :

"Our rectified flow achieves lower reconstruction error for real samples than the original rectified flow."

We decided to remove L141 to avoid redundancy and confusion.
Thank you for pointing this out.

$**[W11]**$
Thank you for raising your concern.
Range of $p_{t}(u)$ is also $[0,1]$. For further details, a visualization of $p_{t}(u)$ is provided in Appendix A.
We hope this clarifies your question.

(to be continued)

$**[W12]**$
We apologize for the confusion.
The scheduling of Slerp depends on $\zeta \in [0, 1]$ --not depend on $t$, and when the maximum $\zeta_{max}$ is fixed (e.g., 0.006), it decreases towards 0 as the training step progresse( Fig 5 ).

I apologize for the confusion caused by representing the Slerp schedule in terms of $t$. The corrected is provided in Revision PDF L301 :

• $\text{slerp schedule}(\zeta) \propto 1 - \frac{{\zeta}^2}{1 + {\zeta}^2}, \zeta \in [0, 1].$

The maximum noise $\zeta_{max}$ is simply scaled linearly, increasing from 0.006 to 0.13 and then decreasing. For example, $\zeta_{max}^{1} = 0.006 \times 1$ to $\zeta_{max}^{4} = 0.13 \times 1$ (Fig 6).
We have incorporated the following revisions into the updated PDF for clarification:

• Lines 298–299: Added explanation for Figure 5.
• Lines 305–308: Added explanation for Figure 6.
• Figures 5 and 6: Adjusted maximum noise to align with the values in the paper.
  We apologize for the oversight and appreciate your feedback

$**[W13]**$
Thank you for raising your curiosity.
This setting fixes the real pair just once and does not continuously update real pairs through repairing.

The ablation study shows that continuously updating real pairs achieves better generation quality than a single pair because it helps preserve the velocity field of the rectified flow for real images.

To provide a more detailed explanation of the ablation settings, we have moved this explanation from Appendix A to the main paper in Lines 504–507.

Changed its name to avoid ambiguity:

• Single Real Pair => Fixed Real Pair

$**[W14]**$
Thank you for raising your curiosity.
Table 1 implies that :

• Better quality in all steps.
• Outperforms even with lower NFE => Indicates reduced drift away from real images in the reflow process.
• Superior with the same distillation method.

Added further explanations in the updated PDF (Lines 350–352), stating:

"Furthermore, our method achieves better generation quality in RK sampling, despite having lower NFE than the baseline. This indicates reduced drift away from real images in the reflow process.”

We appreciate your valuable feedback.

$**[W15]**$
Thank you for your question.
Although the IS values show little difference:

• FID significantly improves with Slerp over no-Slerp.
• Lower $L_2^{\text{recon}}(X_1)$ and $L_2^{\text{p-recon}}(X_1)$ than no-Slerp settings.

This shows the benefits of Slerp.
Unfortunately, generating 4M fake pairs and retraining the rectified flow on another dataset is infeasible within the discussion period. We apologize for the limitation.

$**[W16]**$
Thank you for your suggestion.
The 2-rectified flow trained on the LSUN dataset used only 120k fake pairs for training. This is a significantly smaller number of fake images in the original proposal. Fair FID comparison with the baseline requires at least 4M LSUN fake pairs, which is not feasible within the discussion period.
Instead, we show that our method is significantly better than the baseline in various quantitative metrics :

• IVD(Initial Velocity Delta)
• $L_2^{recon}$ and $L_2^{p-recon}$ of real images.
• $L_2^{recon}$ and $L_2^{p-recon}$ of fake images.
  Table A1 in Appendix C.

Thank you once again for the great interest in our work and the valuable suggestions. If there are any further thoughts or suggestions, they would be greatly appreciated.