Balanced conic rectified flow
Image generation;generative model;rectified flow;diffusion;optimal transport;curvature;ODE
Yes
No ethics review needed.
- In Table 3, why does NFE between the original rectified flow and the proposed method differ (like 110 vs. 104)? 
- Did authors try using only real pairs for reflow (without fake pairs)?
- To the best of my knowledge, using real data and its reverse Gaussian random noise is an interesting and under-explored direction.
- The performance improvement on CIFAR-10 is promising.
- The observations (Figures 2 and 3) that motivate the method are interesting.
This paper presents Balanced conic rectified flow, an improved version of "reflow" in rectified flow. The main idea is to use a real sample and its reverse Gaussian random noise pair based on inverse flow rather than using noise and corresponding generated samples using PF ODE. To mitigate the error from the inverse flow, the paper presents a "conic" reflow that considers a neighbor of the inverse point with a small noise. The paper argues the proposed method shows better $k$-rectified flow for $k>1$, especially for 1-step generation.
I am slightly negative to the paper due to the clarity/readability problem. Specifically:

- The writing of the paper can be improved. For instance, since the primary focus of the paper is the transport from Gaussian distribution to the target data distribution, explicitly mentioning it (like $X_1$ or $X_0$ becomes the real data and the Gaussian noise, respectively), can help the readers to understand the contents better, especially for the people who are not very familiar with the concept of Rectified Flow. Moreover, some captions are overlapped with other figures, such as in Figures 2 and 3. Other figures are really problematic; for instance, in Figure 9 and 10, legends and labels are almost impossible to read. 
-  The paper compares the performance only with the original rectified flow and does not provide a comparison with relevant work [1]. Although it may be understandable because the primary focus of this paper is in revisiting the "reflow" process, I think the quantitative comparison should exist because this work also deals with the same problem. Otherwise, could authors try conic reflow upon this paper? 
- As mentioned in the paper, the paper lacks theoretical analysis when choosing slerp or conic hyperparameters. In this respect I expect more empirical results (like imagenet or various higher-resolution datasets) should be provided, but the in the current form the paper only provides retuls on CIFAR-10 and (qualitative) evaluation on LSUN. Does a similar trend hold for AFHQ or other datasets? I think the paper should include the results at least on all datasets used in the original rectified flow datasets, given that the original paper provides all of the checkpoints for those datasets and this paper does not have theoretical insights (meaning that the validation should be done in empirical manner). 
- The values in Table 2 are difficult to understand. What do they mean?
- Suggestion: do we need to write the reverse ODE as $v^{-1}$? Isn't it just $-v$?
- Minor: curvature -> Curvature in L410.

[1] Lee et al., 2024  Improving the Training of Rectified Flow
Yes
No ethics review needed.
See Weakness
The authors did analysis investigating the rewiring directions in the original rectified flow. There might be a distribution drifting phenomenon that we should pay attention to.
This paper claimed that rewiring the flow starting from the source distribution (e.g. the Gaussian distribution) to the real data distribution in rectified flow generates the drifted distribution. To remedy this, the authors proposed to start from the real data distribution for rewiring. After obtaining the inversion of each real data, the authors use the spherical linear interpolation (slerp) between the reverse noise and a randomly sampled noise. The training objective involves data from both inversion directions.
The authors make some claims without strong evidence:
- The authors mentioned in line 124: "Interestingly, the k-rectified flow underperforms the (k − 1)-rectified flow in terms of image quality. This is obvious because the fake samples have lower quality than the real samples". This sounds too strong. Won't there be other reasons? 
- The authors mentioned that "Since the reflow process uses only fake pairs, discrepancies occur between the reconstruction errors of real and generated images." However, it is still not obvious why reflow process uses only fake pairs is a reason for "discrepancies occur between the reconstruction errors of real and generated images"
- Why "$L^{recon}_2$ is lower at the fake samples than the real samples" indicates "the 2-rectified flow drifts away from the real samples" (line 151)? 
- Why "Lp-recon is lower near the fake samples than the real samples" indicates "the 2-rectified flow suffer from crossing between real samples"  (line 152)?
- The authors mentioned that the original Rectified Flow algorithm (Liu et al. 2022) uses the fake pairs, but I have to mention that the original Rectified Flow algorithm provided an option to use the real data pairs $(v^{-1}(X_1), X_1)$.

Experimental results contradict with the claims:
- From the experiments in Fig. 3(b), the perturbed reconstruction error for real samples are higher than the reconstruction error for real samples. This is not faithful. This contradicts with the perturbation proposed in the method in Sec. 3.3
- This paper discussed that in the original rectified flow, more number of rectified flow lowers the image quality. Therefore, the authors proposed to use real pairs. However the proposed method also suffers from more number of rectified flows being worse (Figure 7). 

Many parts in this paper is not clear and confusing:
- The generated pair or fake pair in this paper seems confusing. Please have formal definitions of the generated pair, fake pair, real pair. 
- What are fake samples? Are those data generated from the source distribution (e.g., Gaussian distribution) via a trained $v$? 
- In line 141 the authors wrote: "Lower reconstruction error on the real samples $_2^{recon}(X_1)$ indicates that a rectified flow is more faithful to the real samples as shown in Figure 3b." However, in Fig. 3(b), real samples have higher reconstruction error. I got confused about what the authors want to express. 
- Definition of exp() in Eq. (10) and (12). Is it an exponential function? I understand $t$ is generally between 0 and 1. 
- In line 295, this method needs to define a slerp schedule with respect to time t. How is the time t in line 295 related to training steps in Fig. 5? How to set the proportional coefficient for the slerp noise schedule? 
- What does the "single real pair" in "reflow using just a single real pair" in Sec. 4.6 Ablation study refer to?
- Table 1's results are not explained. 

Experimental results are not strong:
- From the ablation study, the method without slerp noise performance is similar to the proposed method, in terms of Inception Score, 8.79 vs 8.57. The authors should do experiments on more datasets to evaluate the effectiveness of the proposed real pair and slerp noise.
- No quantitative comparison between the proposed method and the baseline method on the LSUN bedroom dataset. In terms of the qualitative results, when the number of steps is 2 (Fig. A27) or higher (Fig. A28), the results of the proposed method and the baseline method appear close.
Yes
No ethics review needed.
Authors should address the questions raised in Weakness section.
The authors begin with a thoughtful analysis of the limitations in existing rectified flow models, which effectively motivates their proposed approach. Their decision to incorporate real images into the reflow process is both intuitive and straightforward to implement. Additionally, they smartly apply spherical interpolation to blend Gaussian and inverted noise, promoting stable training, and introduce balancing techniques to integrate synthetic-only and real-data flows for greater consistency. The introduction of the IVD metric further enhances the paper’s novelty by offering an innovative tool for in-depth analysis of one-step denoising, providing valuable insights into model behavior and effectiveness, especially in assessing the initial accuracy of generative paths.
The paper begins by identifying two key issues with current rectified flow models: (1) high computational costs due to the large synthetic datasets required for the reflow process, and (2) degradation in image quality with k-rectified flow, where generated images drift from the real data distribution, potentially leading to potential mode collapse. To address these challenges, the authors propose incorporating real images into the k-rectified flow process. This integration significantly reduces computational costs, as fewer samples are needed when using real images (by their experiments). The authors construct noise-image pairs by inverting real images and stabilize the process by blending the inverted and Gaussian noise through spherical interpolation.

Additionally, the paper introduces several enhancements, including balancing between traditional (synthetic-only) and real image-based rectified flows and implementing a Slerp noise schedule. They also propose a new metric, Initial Velocity Delta (IVD), to analyze model performance in one-step denoising. The proposed method demonstrates improvements over traditional k-rectified flow models, as shown through evaluations on CIFAR-10.
Although the paper presents substantial improvements to the k-rectified flow process, a more in-depth theoretical foundation could enhance the rigor and clarity of the contributions. Some key points to consider:
1. The use of real images in reflow may seem to contradict the original purpose of the rectified flow method. Since reflow is based on the fact that it used the new synthesized data which was rewired, making it easier to straighten the path. Real data does not carry that property, so integrating real images might theoretically undermine the reflow process. While experimental results indicate positive outcomes, a theoretical examination of why real images enhance rather than disrupt the reflow process would strengthen the study and clarify this counterintuitive result.
2. The choice of spherical interpolation (Slerp) lacks adequate justification. It would be helpful to understand why Slerp was selected over simpler alternatives like linear interpolation. Is there a theoretical or empirical reason why Slerp provides a significant advantage for stability or image quality in this context? A comparison with other interpolation methods could demonstrate whether Slerp offers unique benefits.
3. Similarly, the noise scheduler’s reliance on spherical interpolation, with a gradual increase and then decrease in Gaussian noise, is not fully explained. An explanation for this scheduling pattern, ideally supported by theoretical or experimental evidence, would clarify its role in enhancing training or image quality and differentiate it from simpler approaches.
4. Since the method stated that it should helps the model avoiding mode collapse, it would be nice to see even more extreme k (papers examined till k=3) to see the effectiveness of the method.
Yes
No ethics review needed.
If the pretrained diffusion model will generate a data samples whose distribution has a drift from real data distribution, it will probably fail to reconstruct the noise with real  samples? That is to say, you can never really achieve a perfect coupling noise and sample with a spoiled pretrained diffusion model. How to handle these situations?
Overall I think the paper is good and reading this paper inspire me to some extent,
1. The motivation is clear (as indicated in the abstracts and introductions) and the overall storytelling is smooth for me.
2. The demonstration is good (good figures and tables).
3. The method is straightforward and easy to understand and implement. 
4. The performance seems to improve a lot compared to the baseline method.
This paper proposes to introduce real samples and their inverted noises (real sample -> noise) instead of commonly used Gaussian noises and generated samples (noise -> generated samples) for rectification (reflow). This operation alleviates the target distribution drift and improve the performance and efficiency of rectification.
1. **Performance**: Although the improvements from vanilla rectified flow, the performance still seems to be  not competitive with advanced distillation and acceleration techniques like iCT [1] and  SiD [2]. For example, in table 1, we can see that the proposed method with rf-3 achieves FID 5.48 (4.68), while in iCT achieves 1-step FID 2.83, 2.51 on CIFAR-10.

2. **Comparison**: For all tables, I would like the authors to add the performance of the original pretrained diffusion models. This will give the readers a clear understanding what is the upper-bound of reflow. Otherwise, readers might find it struggle to understand gap between a pretrained diffusion model  and a trained rectified model. 

3. **Metrics**: I would like to see more metrics like recall and precision to better understand the gap of real data distribution and the generated data distribution. 

4. **Typos**:  In line 746-747,  there is a `??` ; From line 753 to line 768, the words are horizontally centred in a strange way.


5. **Theoretical Analysis**: Though soundness, no critical theoretical analysis or bounds are provided. I'm not implying that there necessarily is one, but I am suggesting that adopting a rational analytical perspective can significantly enrich the substance of the article.

[1] Improved Techniques for Training Consistency Models. ICLR 2024

[2] Score identity Distillation: Exponentially Fast Distillation of Pretrained Diffusion Models for One-Step Generation. ICML 2024
A rectified flow-based generative model that improves generation quality and straightens ODE solution trajectories by incorporating real images into the reflow training process.
@inproceedings{
anonymous2024balanced,
title={Balanced conic rectified flow},
author={Anonymous},
booktitle={Submitted to The Thirteenth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=ctSjIlYN74},
note={under review}
}
Rectified flow is a generative model that learns smooth transport mappings between two distributions through an ordinary differential equation (ODE). Unlike diffusion-based generative models, which require costly numerical integration of a generative ODE to sample images with state-of-the-art quality, rectified flow uses an iterative process called reflow to learn smooth and straight ODE paths. This allows for relatively simple and efficient generation of high-quality images. However, rectified flow still faces several challenges. 1) The reflow process requires a large number of generative pairs to preserve the target distribution, leading to significant computational costs. 2) Since the model is typically trained using only generated image pairs, its performance heavily depends on the 1-rectified flow model, causing it to become biased towards the generated data.

In this work, we experimentally expose the limitations of the original rectified flow and propose a novel approach that incorporates real images into the training process. By preserving the ODE paths for real images, our method effectively reduces reliance on large amounts of generated data. Instead, we demonstrate that the reflow process can be conducted efficiently using a much smaller set of generated and real images. In CIFAR-10, we achieved significantly better FID scores, not only in one-step generation but also in full-step simulations, while using only $7.2\%$ of the generative pairs compared to the original method. Furthermore, our approach induces straighter paths and avoids saturation on generated images during reflow, leading to more robust ODE learning while preserving the distribution of real images.
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Image generation generative model rectified flow diffusion optimal transport curvature ODE
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Balanced conic rectified flow
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