Thank you for recognizing the effectiveness and novelty of our approach, as well as for appreciating the comprehensiveness of our experiments and the practical relevance of addressing distribution drift in ReFlow.

W1,2

Thank you for this insightful comment. You are absolutely right: in this context, $v(Z_0)$ is not meant to represent the velocity field evaluated at a single point, but rather the outcome of solving the ODE defined by the velocity field. In other words, our intention was to represent the trajectory mapping $T(Z_0) := \int_0^1 v(z_t, t) dt$, and we used the notation $v(Z_0)$ as a shorthand for this operator.

To simplify notation and avoid introducing an additional operator like $T$, we opted for this form. However, as you point out, this may lead to confusion, especially when integration is not explicitly stated.

We also acknowledge that in Footnote 1, the integral form was unintentionally omitted, which may have compounded this ambiguity. We will revise the footnote to explicitly include the integral definition, and we will consider introducing auxiliary notation (e.g., $T(Z_0)$) in the revised version to improve clarity and prevent confusion for future readers.

We sincerely appreciate your careful reading and helpful suggestion.

W3

Thank you for pointing this out. Indeed, $\epsilon$ and $\zeta$ are sampled during training and introduce stochasticity. However, in Equation (9), we focused on clearly conveying the core structure of the conic reflow loss. For simplicity, we presented the expectation over time $t$, treating $\epsilon$ and $\zeta$ as fixed samples per iteration.

While it is more mathematically precise to express the full expectation over all random variables $\epsilon$, $\zeta$,and $t$, omitting some expectations is a common simplification in generative modeling literature, especially when the omitted variables are re-sampled at every training step.

That said, we appreciate the reviewer’s suggestion and will denote the sampling of $\epsilon$ and $\zeta$ more explicit in the final version for improved clarity and rigor.

W4

We identify a meaningful gap between:

• Reconstruction error:
  $L_{\text{recon}}^2(X) = \mathbb{E}_{x \sim \mathcal{X}} \| x - v(v^{-1}(x)) \|^2$
• Perturbed reconstruction error:
  $L_{\text{p-recon}}^2(X, \varepsilon) = \mathbb{E}_{x \sim \mathcal{X}, z \sim \pi_0} \| x - v(v^{-1}(x) + \varepsilon z) \|^2$

We argue that Conic ReFlow effectively mitigates drift (i.e., bias toward fake samples), and provide the following rationale:

1. Generative models inherently cannot perfectly learn the ideal target distribution.
2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.
3. A noticeable gap in reconstruction and perturbed reconstruction error is observed between fake and real images.
4. This gap indicates that the model is biased toward fake images and lacks robustness around real samples.
5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.
6. As a result, it narrows the real/fake error gap and alleviates degradation in full-step quality.

Conic Reflow is theoretically sound and geometrically motivated. Importantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].

Our method learns not just to map a point but to cover the cone-shaped neighborhood around real data on the manifold. This enables training of geometry-aware flows that remain robust in real data regions—something not achievable by straight paths generated from fake-only inversions.

[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations
[2] Anomaly Detection with Robust Deep Autoencoders
[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection
[4] Šmídl et al., Anomaly Scores for Generative Models (2019)
[5] Adversarial Examples for Generative Models, ICLR Workshop (2020)
[6] Input Perturbation Reduces Exposure Bias in Diffusion Models (arXiv:2404.01734)
[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection
[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning (NeurIPS 2018)
[9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.

W5

1. The direct performance difference between using only real+fake data ([2]) and applying Slerp can be seen in our ablation study (Section 4.4, Table 2).
   The comparison between** "No-Slerp"** and "Ours" clearly demonstrates the impact of Slerp-based perturbation.

Strengths of Ours

1.Better generation quality.
2.Straighter sampling path.
3.Lower reconstruction and perturbed reconstruction error on real images.

Using only real pairs is not sufficient

1. Despite worse generation quality, IVD is lower.
   • This indicates that the model initially predicts a worse velocity direction compared to ours.
   • In other words, simply using real pairs does not sufficiently mitigate the bias toward fake samples in standard ReFlow.

2.[2] analyzes the self-consuming training issue by measuring the magnitude of DAE model weights.

This is a perspective focused on the training dynamics. In contrast, we target the distributional drift and reconstruction error directly and propose a perturbation-aware conic guidance mechanism. Thus, although both methods aim to improve reflow, our approachs are trajectory and bias toward fake samples, and are fundamentally different from [2].

[2] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.

Q1

We appreciate the detailed feedback.

• In the final version, we will unify the notation for k-rectified flow using consistent LaTeX math style ($k$) throughout the text.

Q2

Thank you for the detailed feedback.

• The set $\mathbb{N}$ refers to the total number of training steps involving both real and fake pairs (Lines 159–160).
  However, we acknowledge that the reference in Line 161 to Section 3.3 for its definition may cause confusion.
  To avoid confusion, we will remove the notation $\mathbb{N}$ from the corresponding line in the final version.

• Equation (11) presents the overall loss function that combines both fake and real pair supervision.

• Equation (12) provides the indexing rule that determines which of the two loss terms is active at each training iteration.

Specifically:

• If the training index $i \in U_{\text{fake}}$, then the first term $\chi_{\text{fake}} \cdot \left( \dot{Z}_{t,F} - v_\theta(Z_{t,F}) \right)^2$ is activated.
• If $i \in U_{\text{real}}$, then the second term $\chi_{\text{real}} \cdot \left( X_1 - \text{slerp}(Z_{0,R}, \epsilon, \zeta) - v_\theta(Z_t^{\text{Conic}}) \right)^2$ is used.

Therefore, Equation (12) is not an empirical approximation of (11), but rather describes the index set over which the individual terms in (11) are selected and applied during training.

We hope these clarifications help resolve the reviewer’s concerns We hope this addresses your concern regarding computational efficiency, and we would be glad to provide further clarification during the discussion phase.

Thank you for clarifying this. I am okay if the suggested changes are implemented in the revised version. I have increased my score.

We appreciate the reviewer for positively noting our ablation studies, potential applicability to larger flow models.

W1

We thank the reviewer for the detailed comparison. However, we respectfully argue that the evaluation that Conic ReFlow underperforms compared to [2] and [3] is not fully justified for the following reasons.

While the results reported in the main paper of [2] are based on a different configuration with batch size 512 and 800K iterations, we apply Conic ReFlow to RF++ under the same setting as the original Rectified Flow, using batch size 128 and 500K iterations, to ensure a fair comparison.

As shown in Table 1 and Table 6 in Appendix G, our method achieves better FID and IS scores under this aligned setting.

• Same configurations EDM initialization, LPIPS-Huber-1/tloss, batch size 128 with 500K training itration.
• Different configurations: RF++ / RF++(+Ours) uses 800K/600K fake pairs.

Notably, even with fewer training pairs, our method outperforms the baseline, highlighting that Conic ReFlow consistently improves upon standard reflow even when applied to stronger flow models like [2].

Conic ReFlow is not intended to compete with existing rectified flow techniques but rather to offer a complementary improvement. Our primary objective is to alleviate the drift issue that arises during the reflow process.

In contrast, [2] and [3] focus on overall generation performance improvements, such as EDM initialization, refined loss, and better samplers. Conic ReFlow provides a structural contribution by improving the stability during the reflow.

Therefore, to ensure a fair comparison, all experiments in our paper are conducted using Rectified Flow as the baseline initialization. In addition, the reported results on ImageNet are based on unconditional generation, so comparing image quality against class-conditional generation, which benefits from explicit guidance, is not appropriate.

We also clarify that the purpose of Table 4 and 5 is to empirically demonstrate that our proposed Conic ReFlow consistently improves overall generation quality (FID, IS, and recall) over the original ReFlow step when applied to the same baseline, particularly on more complex datasets. It is not intended as a direct comparison against current state-of-the-art flow models.

Our contribution goes beyond merely improving FID scores. The primary goal of Conic ReFlow is to evaluate and enhance the straightness of the reflow path, diversity, fidelity, and mitigation of drift. To this end, we employ a variety of metrics:

• Straightness, IVD
• Precision / Recall
• $L_{2}^{recon}, L_{2}^{p-recon}$ and these gaps

Our method demonstrates consistent improvements across diverse datasets, including CIFAR-10, ImageNet 64x64, and LSUN.

In fact, the most methodologically similar approach to ours is SlimFlow [1], which also employs an Annealing ReFlow strategy using noise interpolation for performance gains:

Conic ReFlow is a novel approach to structurally address the inherent issues of the ReFlow process. This contribution is complementary rather than competitive to prior methods, and its validity is supported by extensive experimental evidence.

[1] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.
[2] Improving the Training of Rectified Flows, NeurIPS, 2024.
[3] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.

W2

We thank the reviewer for their detailed and thoughtful questions.

Our claim is that discontinuities near the data manifold can indicate potential global drift. We do not claim that continuity is theoretically a sufficient condition for global alignment.

The connection between the local continuity and global continuity stems from the structural nature of flow models, where the global vector field is gradually constructed through the accumulation of conditional path supervision [1]. Specifically, recent work leverage local continuity as a means to improve global performance, including generalization guarantees under locally Lipschitz conditions [2, 8]. It is a commonly accepted view that local behavior can influence the overall quality of generation.

W3

Thank you for the valuable comment. We have quantitatively demonstrated that Conic ReFlow mitigates drift (i.e., bias toward fake images), and this is supported by the following evidence:

1. Generative models inherently cannot perfectly learn the ideal target distribution.
2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.
3. A noticeable gap in reconand p-recon error is observed between fake and real images.
4. This gap indicates that the model is biased toward fake images and lacks robustness around real samples.
5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.
6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.

Importantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [3–7], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [8–10].

[1] Flow Matching for Generative Modeling
[2] De Bortoli et al., Convergence and Generalization of Score-Based Generative Models*(2023)
[3] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations, CoRR, 2018
[4] Anomaly Detection with Robust Deep Autoencoders, KDD 2017
[5] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection, BMVC 2015
[6] Šmídl et al., Anomaly Scores for Generative Models, arXiv:1905.11890, 2019
[7] Adversarial Examples for Generative Models, ICLR Workshop, 2020
[8] Input Perturbation Reduces Exposure Bias in Diffusion Models, ICML, 2023\ [9] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection, CVPR, 2024
[10] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning (NeurIPS 2018), NeurIPS, 2018\

W4

We justify our design choice based on both intuitive considerations and theoretical motivations from recent literature:

1. Reverse noise are inherently imperfect : These pairs are generated through imperfect inversion processes. If reused without refresh, they can cause the model to overfit or saturate to suboptimal or outdated couplings.

2. Motivation from Minibatch and Multisample Coupling Strategies : As shown in [1] and [2], jointly constructed couplings over minibatches or multisamples better approximate ideal transport plans than independent pairings. These methods explicitly construct $\pi(x,y)$ on a batch to reduce training variance and improve flow straightness and transport cost.

3. Bias mitigation and perturbation-aware stability : Periodic refreshing prevents bias from stale supervision and ensures that reverse noise perturbations act on diverse, up-to-date pairs—promoting stability and robustness during training.

[1] Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport, TMLR , 2024.
[2] Multisample Flow Matching: Straightening Flows with Minibatch Couplings, PMLR , 2023.

W5

We respectfully clarify that while our method and Slim Flow [1] both involve perturbed supervision, the motivation, design intent, and experimental behavior differ significantly:

Slim Flow introduces Annealing Reflow solely to provide an efficient initialization for a smaller student model by gradually transitioning the target velocity from random to teacher model.

In contrast, our Conic Reflow is designed to improve the reflow process itself, by correcting biases introduced by imperfect real noise couplings. Our method is applicable to models of any size and is not limited to distillation settings.

Slim Flow only evaluates 1-step generation quality, as its objective is to distill a teacher model's immediate behavior. It does not report or ensure performance on multi-step sampling.

In contrast, our method directly improves both few-step and multi-step generation quality (see Table 1), making it robust for practical generative tasks.

Slim Flow lacks any data-driven or path-aware adaptation of the annealing parameter $\beta$.

Conversely, our Conic Reflow includes an adaptive $\zeta_{\max}$ formulation (Eq. 13) that adjusts perturbation intensity, providing more principled and effective supervision.

W6

While both methods utilize real data points, the motivations and problem setups are fundamentally different.
Simple Reflow employs forward pairs as a minimal remedy to mitigate error accumulation.
In contrast, our method aims to address a more foundational issue by leveraging real and generated image pairs.
Specifically, we mitigate instability and bias in the reflow process, construct adaptive paths, and enhance training stability.

Q1

• Clarity :
  Through responses to W2 and W3, we provided logical explanations, experiments, and supporting evidence.
• Originality :
  Through W5 and W6, we clarified the differences between our method and proper baselines, highlighting the originalit of our approach.
• Significance:
  In W1, we clarified that our experiments include a direct comparison under the same configuration.

We hope these clarifications help resolve the reviewer’s concerns and would be glad to provide further clarification during the discussion phase.

Dear Reviewer,

This is a gentle reminder that the Author–Reviewer Discussion phase ends within just three days (by August 6). Please take a moment to read the authors’ rebuttal thoroughly and engage in the discussion. Ideally, every reviewer will respond so the authors know their rebuttal has been seen and considered.

Thank you for your prompt participation!

Best regards,

AC

Thank you for the thorough rebuttal. The rebuttal has partially addressed my concerns with

• Significance [W1] with experiments on CIFAR10 (but experiments on ImageNet is missing),
• Clarity [W1], [W2], [W3] with further discussion.

However, some concerns with Originality [W1], [W2] still remain unaddressed:

• No numerical comparisons to relevant baselines such as noise annealing in SlimFlow and real-fake pair balancing in Simple ReFlow,
• Incremental nature of the proposed method compared to previous works.

Hence, I have raised the score to borderline reject.

Thank you for your updated comment, and we’re glad that our previous rebuttal has helped improve clarity.

We agree that including comparisons on ImageNet between RF++[1] and RF++(+ours), as well as applying our method to the models of [2]and [3], would further strengthen the persuasiveness of our contributions. However, we would like to respectfully explain that conducting additional from-scratch experiments involving [2] and [3] within the discussion period is realistically infeasible due to the following reasons:

• To perform ReFlow, we must generate at least 1M fake pairs for CIFAR10 and ImageNet, respectively.
• For a fair comparison, we would need to train [1], [2] and [3] with our methods from scratch with their full settings:
  • [1] uses batch size 2048 with 700K iterations on ImageNet64.
  • [2] uses batch size 512 with 200K iterations on CIFAR10 and batch size 1024 with 500K iterations on ImageNet64.
  • [3] uses 1.2M iterations total with an unknown batch size in annealing reflow process.

Unfortunately, given our limited GPU resources, this level of training is not realistically possible within the rebuttal period.

As a feasible alternative, we instead focused on clearly articulating the differences in motivation, problem formulation, key contributions, and methodological design between our method and those in [2,3].

Our method tackles the drift phenomenon during ReFlow by introducing perturbation-based supervision on conic paths around real samples. This is fundamentally distinct from the noise interpolation of [2] or the fixed real-fake balancing of [1], both in motivation and design.

Empirically, we also note that SlimFlow’s ablation Table 3 reports an RK-step FID of 5.46 with annealing ReFlow (NFE unspecified), while our RK-step FID reaches 3.24.
More notably, our 1-step distilled model achieves FID 4.16, surpassing even the RK-solver quality of SlimFlow’s annealing ReFlow baseline.

In addition, our Conic ReFlow offers a general and geometry-aware perspective that may help explain why annealing ReFlow empirically works well.

While SlimFlow [3] introduces an annealing schedule, the choice of its key hyperparameter $\beta$ is not fully justified. In contrast, our $\zeta_{\text{max}}$ is computed adaptively based on the learned velocity field and the training data distribution, as described in Equation (13). This principled formulation suggests that Conic ReFlow has the potential to generalize better across datasets.

We sincerely hope that our response has clarified the originality and focused contribution of our method. We greatly value your suggestion, and to further address your concern, we plan to include additional empirical comparisons in the camera-ready version, where we will apply our Conic ReFlow to [2] and [3] and report generation quality for stronger support.

Once again, thank you for acknowledging the improvement in clarity. We hope that this response further helps address the concerns regarding originality, and we’re happy to provide any further clarifications during the discussion phase.

[1] Improving the Training of Rectified Flows, NeurIPS, 2024.

[2] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.

[3] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.

We thank the reviewer for their careful reading and appreciation of our key ideas, including the conic reflow design, drift mitigation, and strong empirical performance across datasets.

W1

Thank you for the valuable comment.

While we do not provide a formal theoretical analysis, Conic ReFlow is logically designed to mitigate drift (i.e, biased toward fake images). Its effectiveness is supported by empirical results, and its rationale can be reasonably grounded in existing literature on reconstruction-based robustness and reduces Bias.

1. Generative models inherently cannot perfectly learn the ideal target distribution.
2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.
3. A noticeable gap in reconand p-recon error is observed between fake and real images.
4. This gap indicates that the model is biased toward fake images and lacks robustness around real samples.
5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.
6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.

Importantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].

[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations, CoRR, abs/1802.00187, 2018
[2] Anomaly Detection with Robust Deep Autoencoders, KDD 2017
[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection, BMVC 2015
[4] Anomaly Scores for Generative Model, arXiv:1905.11890, 2019
[5] Adversarial Examples for Generative Models, ICLR Workshop, 2020
[6] Input Perturbation Reduces Exposure Bias in Diffusion Models ICML, 2023
[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection, CVPR, 2024
[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning, NeurIPS, 2018
[9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024

W2

Thank you for the detailed feedback.

Our paper includes comparisons with recent diffusion-based models in Appendix M to further contextualize the performance of reflow-based methods. It is mentioned in L210 and Section 4.6.

We suggest that including comparisons with more recently developed flow-based generative models would help readers better understand the overall generative quality landscape. We will provide FID and IS scores of recent flow models, including [1–4], in Appendix M of the final version.

[1] Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport, TMLR , 2024.
[2] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.
[3] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.
[4] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.\

W3

We appreciate the reviewer’s request for additional high-resolution experiments.

We would like to note that we have already included high-resolution results on the LSUN Bedroom dataset in our current submission, covering FID, precision, recall, $L_2^{recon}$, and $L_2^{p\text{-}recon}$ (see Figure 8). We also provide qualitative comparisons across random seeds in Appendix (Figures A6–A10), demonstrating the robustness and effectiveness of Conic ReFlow.

However, further experiments on other high-resolution datasets such as LSUN-Church, AFHQ, or ImageNet 256 are realistically infeasible within the rebuttal period due to the following constraints:

1. The original Rectified Flow repository only provides 1-Rectified Flow checkpoints for these datasets, requiring us to re-train 2-Rectified Flow baselines from scratch for fair comparison.

2. High-resolution datasets of this scale require at least 120K fake samples to produce meaningful comparisons under the reflow paradigm.

3. Given our limited GPU resources, training with batch size 16 would take approximately 4 weeks to reach convergence and obtain statistically meaningful results.

We sincerely regret that we are unable to include these additional results in the rebuttal. However,** we plan to report full evaluations on LSUN-Church and AFHQ datasets in the camera-ready version**, including comparison with the original Rectified Flow and release of corresponding model checkpoints.

We hope this addresses your concerns and clarifies the practical limitations involved.

Q2

We appreciate the reviewer’s concern regarding hyperparameter sensitivity.

We provide several ablation studies on different Slerp settings.

For Slerp perturbation patterns, we provide detailed comparison in Table 3, showing that the interpolation method yields consistent quality across different variants.

• Ours Slerp Pattern: $\zeta_t=\zeta_{\text{max}}/K \rightarrow\zeta_{\text{max}} \rightarrow \zeta_{\text{max}}/K$
• Strictly Increasing:$\zeta_{t}=\zeta_{\text{max}}/K \rightarrow\zeta_{\text{max}}$
• Strictly Decreasing: $\zeta_{t}=\zeta_{\text{max}}\rightarrow\zeta_{\text{max}}/K$

Futhermore, appendix E Table A4 compares the IS/FID performance when using only real pairs, using both real and fake pairs, and applying our conic reflow method.

• Real-only pairs (Original Rectified flow using only Real pair),
• Real + Slerp-perturbed pairs,
• Real + Fake + Slerp pairs

Overall, the results empirically support that conic reflow consistently outperforms standard reflow regardless of the presence of fake pairs or the specific noise injection pattern used.

2. $\zeta_{max}$ is adaptively selected

We would like to clarify that: $\zeta_{max}$ is not manually tuned but adaptively selected based on the maximum p-reconstruction error between fake and real samples (eqn 13 in Sec3.5).

We acknowledge that Table 3 may have caused confusion by presenting a fixed value for $\zeta_{\max}$. In the final version, we will revise this to clarify the scheduling pattern:

• For strictly increasing : $\zeta_{t}=\zeta_{\text{max}}/K \rightarrow\zeta_{\text{max}}$
• For strictly decreasing : $\zeta_{t}=\zeta_{\text{max}}\rightarrow\zeta_{\text{max}}/K$

Once again, thank you for the helpful suggestion, and we hope this response has clarified your concerns.

Q3

Thank you for the thoughtful question.

We would like to clarify that the computational cost of conic reflow is essentially the same as that of the original rectified flow, except for the initial step of estimating $\zeta_{\max}$.

Importantly, conic reflow is significantly more efficient in terms of total training resources. Unlike the baseline our method achieves strong performance using only about 7–10% of fake pairs. As a result, the overall training cost is actually lower than that of the baseline.

We hope this addresses your concern regarding computational efficiency, and we would be glad to provide further clarification during the discussion phase.

Dear Reviewer,

This is a gentle reminder that the Author–Reviewer Discussion phase ends within just three days (by August 6). Please take a moment to read the authors’ rebuttal thoroughly and engage in the discussion. Ideally, every reviewer will respond so the authors know their rebuttal has been seen and considered.

Thank you for your prompt participation!

Best regards,

AC

We appreciate the reviewer’s agreement with our motivation and contributions, especially the drift analysis and improvements over existing reflow methods.

W1

We appreciate the reviewer’s concern and the opportunity to clarify our choice of analysis. We suggest that 1-step reconstruction is an appropriate and informative diagnostic for drift (i.e., bias), for the following reasons:

1. Support from prior work: Prior studies on exposure bias in diffusion models (e.g., [1], [2]) emphasize that the early steps of sampling are where the most critical errors or biases arise. Our use of 1-step evaluation is consistent with this literature.

2. Reduced solver accumulation error: Multi-step sampling introduces numerical errors that accumulate with each step, potentially masking model-induced error. 1- or 2-step evaluation avoids this confounding factor and more cleanly isolates the model’s contribution to reconstruction performance.

3. Direct signal of drift (bias toward fake samples): Since the reflow trajectory heavily depends on the model’s predicted initial velocity, a 1-step Euler reconstruction error directly reflects the alignment of the model’s estimated vector field with the true reverse direction.

4. Statistical robustness: We report reconstruction errors averaged over repeated evaluations on 10k samples using the same solver settings. The consistent error gap between real and fake samples likely reflects intrinsic model behavior, not solver stochasticity.

5. Multi-step support in supplement: In Supplementary Figure 5, we also report 2-step Euler reconstruction error on ImageNet. The gap between fake and real reconstruction errors remains, and our proposed conic method demonstrably narrows this gap.

[1] Elucidating the Exposure Bias of Diffusion Models ICLR, 2024
[2] Input Perturbation Reduces Exposure Bias in Diffusion Models ICML, 2023

W2

We appreciate the reviewer’s concern regarding hyperparameter sensitivity.

1.We provide several ablation studies on different Slerp settings.

Regarding the mixing between real and fake pairs, we provide an ablation study covering:

• Real-only pairs (Original Rectified flow using only Real pair),
• Real + Slerp-perturbed pairs,
• Real + Fake + Slerp pairs,

demonstrating that performance improves consistently when our full method is applied (Supp. Fig. A4).

For Slerp perturbation patterns, we provide detailed comparison in Table 3, showing that the interpolation method yields consistent quality across different variants.

• Ours Slerp Pattern: $\zeta_t=\zeta_{\text{max}}/K \rightarrow\zeta_{\text{max}} \rightarrow \zeta_{\text{max}}/K$
• Strictly Increasing:$\zeta_{t}=\zeta_{\text{max}}/K \rightarrow\zeta_{\text{max}}$
• Strictly Decreasing: $\zeta_{t}=\zeta_{\text{max}}\rightarrow\zeta_{\text{max}}/K$

2. $\zeta_{max}$ is adaptively selected

We would like to clarify that:

$\zeta_{max}$ is not manually tuned but adaptively selected based on the maximum p-reconstruction error between fake and real samples (eqn 13 in Sec3.5).

To further address the reviewer’s concern, we will additionally report generation quality across varying fake-to-real pair ratios, comparing both the baseline and our method.

Key takeaways:

1. Conic ReFlow improves 1-step performance across all settings.
2. 600K achieves the best trade-off between efficiency and performance.
3. Too few fake pairs (e.g., 60K) results in poor full-step generalization.(For detailed analysis, see Appendix E.)

These results collectively indicate that our approach maintains robust performance across a range of hyperparameter configurations, and the reported improvements are not heavily reliant on specific tuning choices.

We hope this alleviates the reviewer’s concern.

W3

Thank you for the valuable comment.

We argue that Conic ReFlow effectively mitigates drift (i.e., bias toward fake samples), and provide the following rationale:

1. Generative models inherently cannot perfectly learn the ideal target distribution.
2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.
3. A noticeable gap in recon and p-recon error is observed between fake and real images.
4. This gap indicates that the model is biased toward fake images and lacks robustness around real samples.
5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.
6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.

Importantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].

[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations
[2] Anomaly Detection with Robust Deep Autoencoders
[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection
[4] Šmídl et al., Anomaly Scores for Generative Models (2019)
[5] Adversarial Examples for Generative Models, ICLR Workshop (2020)
[6] Input Perturbation Reduces Exposure Bias in Diffusion Models, ICML, 2023
[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection
[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning, NeurIPS, 2018 [9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.

Q1

Thank you for the helpful suggestion.
We acknowledge that the readability of Figures 3a–b and 5b can be improved and will revise the arrangement and legends in the final version as follows to enhance clarity:

• For Figure 3(a), we will move the legend to the top center and increase the font size of both the legend and the values above the bars.
• For Figure 3(b), we will remove the duplicated legend and relocate the remaining one to the top right, increasing the font size to improve readability.

Appreciate your attention to detail.

Q2

Thank you for the helpful suggestion. We agree that the current description is misleading, as it omits the integration process and may imply a direct addition of velocity at a single time step. We will add that integration over time is required in the final version.

We would be happy to elaborate over the discussion phase

We're glad to see that [W1] and [W2] concerns have been addressed, and hope the following theoretical clarification helps resolve your question regarding our perturbation-based supervision.

Fundamental Limitation of Ideal Target distribution Analysis
As the true data distribution is generally unknown in practice, mathematically deriving how close our supervision lies to an ideal target distribution remains a largely open problem. To the best of our knowledge, there exists no comprehensive framework for characterizing this distance in a nonparametric setting.

Theoretical Foundations of Local-to-Global Flow Construction with Perturbed Path Supervision
Flow-based models construct the global vector field gradually through the accumulation of local supervision signals along individual paths [1]. This motivates our use of perturbation-based supervision near real data.

Importantly, recent theoretical developments [2] provide rigorous support for the stability of such localized training when the data distribution $p^\star \in \mathcal{A}_K^\beta$ satisfies mild regularity assumptions (e.g., $\beta$-Hölder smoothness, subGaussian tails).

In particular:

• Proposition 6 guarantees that the score function $s^\star(t, x)$ is uniformly bounded in space and time.
• Proposition 7 establishes one-sided Lipschitz continuity via an exponentially decaying bound on the Jacobian eigenvalues.
• Theorem 8 shows that the score is Hölder smooth of arbitrary order on high-probability subsets, with time-dependent norm control.

These results imply that perturbation paths around real samples—such as those used in our method—lie within well-behaved regions of the score field, allowing stable and informative supervision. As a result, our perturbation-based supervision contributes valid gradient signals for constructing the global vector field, without violating the theoretical assumptions required for convergence.

This perspective assumes that the reverse noise associated with real images also lies on a similarly well-behaved manifold—i.e., it is β-Hölder smooth with subGaussian tails, and its score function is bounded and Lipschitz, as in [2]. Under this view, our perturbation-based supervision operates within regular regions of the score field and aligns well with the theoretical foundations of score-based generative modeling.

Compatibility with Flow Matching Theory
Since diffusion models can be interpreted as special cases of flow models under particular path constraints [3,4], the theoretical results from the flow matching literature are directly relevant to our framework. The robustness of perturbed supervision paths is thus supported by the same regularity arguments used to justify flow matching convergence.

Formulation within Conditional Flow Matching (CFM) Framework
Our perturbation-based approach adheres fully to the conditional flow matching (CFM) formulation proposed in [1]. Specifically, our training objective can be written as:

$\mathcal{L_{CFM}}(\theta) = \mathbb{E}_{t \sim \mathcal{U}[0,1],\ x_1 \sim q(x_1),\ \epsilon \sim \mathcal{N}(0,I)} \left[ \left\| v_t(\hat{x}) - u_t( \hat{x} \mid x_1) \right\|^2 \right]$

where $\hat{x} = slerp(x,\epsilon, \zeta)$ , and consider conditional probability path of the form :

$p_{t}(\hat{x} \mid x_{1}) = \mathcal{N}(\hat{x} \mid \mu_{t}(x_{1}), \sigma_{t}(x_{1})^2I)$

where $\mu$ and $\sigma$ are time-dependent mean and scalar standard deviation respectively. Given the assumptions above, the theory in [1] guarantees that such loss formulation leads to a well-posed and convergent estimation of the global vector field.

In summary, while we do not provide a formal proof that our perturbation-based supervision approximates an ideal target distribution more closely (nor do we claim a new theoretical bound in this work), we do provide theoretical grounding that:

• Our approach operates within the well-established flow matching framework,
• It inherits its convergence and regularity properties under mild and reasonable assumptions,
• And it introduces no violations to the theoretical assumptions required for flow training stability and robustness.

We sincerely hope this explanation alleviates your concerns regarding the theoretical basis of our method. Exploring explicit bounds or optimality conditions for perturbed supervision remains an exciting direction for future work, beyond the current scope of this paper.

We would be happy to further elaborate on any follow-up questions during the discussion phase.

[1] Flow Matching for Generative Modeling, ICLR, 2023.

[2] Convergence and Generalization of Score-Based Generative Models, https://arxiv.org/abs/2507.04794, 2023.

[3] Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow, ICLR, 2023.

[4] Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow, ICLR, 2025.