We thank the reviewer for their positive and constructive review. We address the concerns below.

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Theoretical Claims / 1.[C] Lines 791-793: as I understand, $\dot{\tilde{x}}_t=\dot{\sigma}_t z$?

$\dot{\tilde{x}}_t = \frac{d(\hat{x}_t)}{dt} + \dot{\sigma}_t z \neq \dot{\sigma}_t z$ where $\hat{x}\_t = G\_\theta(x\_t, \sigma\_t)$. However, a pair of training GC points starts from the same $\hat{x}_t$, which sets the velocity term to $\dot \sigma_t z$.

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Experimental Designs Or Analyses / 1.[Q] I do not completely understand the experimental setup in sections 4.2.1 and 4.2.2. What was chosen as an "ideal consistency model”?

The ideal consistency model in all experiments of Sections 4 and 5 is a consistency model trained in a standard manner on independent coupling (iCT-IC), as described in Section 5.1. We apologize for the confusion and will clarify this point in Section 4.

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Experimental Designs Or Analyses / 2.[C] Figures 2, 3, x-axis. What is “timesteps $\sigma$”?

It is the diffusion noise's standard deviation $\sigma_t$, as in $x_t = x_0 + \sigma_t z$.

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Experimental Designs Or Analyses / 3.[I] The gap in $\tilde{R}$’s values in Figure 2 does not seem to be significant. Do we really need to fight for such an improvement?

Note that on large timesteps, there is an order of magnitude of difference between IC and GC. Then, the relationship between improvement in $\tilde{R}$’s values and improvement in final sampling quality is unknown. We hypothesize that a small improvement of $\tilde{R}$ could lead to significant change on sampling quality, as our analysis suggests.

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Experimental Designs Or Analyses / 4.[I] Table 1: In the original work about iCT [Song and Dhariwal, 2024], the authors report FID values ~ 2 smaller than yours. Why such a difference? Also, how many function evaluation steps was used to obtain the FID?

We report FID with 1 NFE. The difference with iCT stems from the following differences: iCT uses 400k iterations with batch size 1024, while we use 100k with batch size 512. For fairness, all models were trained and evaluated in the same setting. Note that our ECT experiments follow the original setting from the authors, and that we present new results (see rebuttal to Reviewer PJfJ).

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Other Strengths And Weaknesses. Does Generator flow worsens stability issues of consistency models?

We observed instabilities in two settings: with mixed precision in the iCT setting; with large pre-trained models in the short training setting of ECT. In both cases, instabilities arose for both IC and GC.

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Other Comments Or Suggestions. I do not understand why the discrepancy between CT and CD losses matters. [...] Why is it an issue that needs to be addressed, as outlined in the abstract and contributions? [...] I expect that the main problem (it is also mentioned in the text) is the variance of $L_{CT}$ estimator. [emphasis ours]

CD provably approximates the target diffusion flow, ensuring convergence to the data distribution. We show that CT is not equivalent to CD due to a Jacobian regularization term (Theorem 1), which prevents CT from learning the true diffusion flow. This challenges widely accepted knowledge and suggests that the generator learned by CT may not converge to the data distribution. We hypothesize that addressing this discrepancy improves performance, as confirmed by our experiments.

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Questions / 1. How do the results of Theorem 1 correspond to those by [Song et. al]. What was lost in the analysis by [Song et. al.] that they “missed” the discrepancy between CT and CD? [emphasis ours]

The difference comes from two issues:

• in their Theorem 2 ($L_{CT} = L_{CD}+o(\Delta T)$), the $o(\Delta T)$ is actually too large compared to the other term, and consequently the result is uninformative;
• their Theorem 6 (limiting gradient equality) is stated with a general distance function, but the requirements on its Hessian restrict the theorem's validity to a quadratic loss.

We will clarify those differences in the next version of the paper.

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Questions / 2. Theorem 1, case $\alpha=2$. How could it be the case that limiting gradients are equal, but the objectives themselves are not, and the difference between objectives depends on $\theta$.

The case $\alpha=2$ is particular: objectives are not equal, and gradients of the limit loss are not equal. However, limiting gradients are equal because of the interplay between the "stop-gradient trick" (whose effect disappears only in continuous time and not in the limit) and the quadratic loss (cf proof of Theorem 1, L668). This equality is in some sense a coincidence.

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Minor / 1. What is $\sigma_d$?

As in EDM (Karras et al., 2022) and later works, $\sigma_d=0.5$ in our experiments.

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Minor / 2. It is noted that $v_t(x) = \mathbf{E}[\dot{x}_t | x_t =x]$. [...] Some references?

This can be seen in Liu et al. (2023), Definition D.1.

We would like to thank the reviewer for their detailed and positive assessment. We address the raised questions/weaknesses below.

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The paper claims that [$\tilde{\mathcal{R}}_t$] is "a proxy term for $\mathcal{R}(\theta)$", [...] but does not provide any mathematical justification for this. [...] I encourage the authors to make this connection clearer. [emphasis ours]

We can make this connection clearer in the case of the quadratic loss ($\alpha=2$). Indeed, we can bound the regularization term with the proxy term thanks to the Jacobian's maximum singular value $s_{\text{max}}( \frac{\partial f_\theta}{\partial x} )$, which is bounded as typical networks are Lipschitz:

$\left\| \frac{\partial f\_\theta}{\partial x}(x\_t, \sigma_t) \left( \dot{x}\_t - v_t(x\_t) \right) \right\|^2 \leq \left\| \frac{\partial f\_\theta}{\partial x} \right\|^2 \|\dot{x}\_t - v\_t(x\_t) \|^2 \leq s^2\_{\text{max}} ( \frac{\partial f\_\theta}{\partial x} ) \|\dot{x}\_t - v\_t(x\_t) \|^2$

Moreover, under some other assumptions on $f$, for example the fact that it is close to a scaling function for large $t$ (see Corrolary 2), if $f(x, \sigma_t) = \frac{\sigma_0}{\sigma_t} x$, then we would have:

$\left\| \frac{\partial f_\theta}{\partial x}(x_t, \sigma_t) \left( \dot{x}_t - v_t(x_t) \right) \right\|^2 = (\frac{\sigma_0}{\sigma_t})^2 \|\dot{x}_t - v_t(x_t) \|^2$.

We will include this discussion in the next version of our paper.

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Typos. We thank the reviewer for all the suggestions and the careful reading. We will modify the paper accordingly.

We would like to thank the reviewer for their thorough and constructive review. We address the raised questions/weaknesses below.

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1. How would you explain that using minibatch optimal transport on LSUN slightly outperforms the generator coupling? [emphasis ours]

One of our contributions is to explain why using batch-OT in consistency models is a strong baseline via our theoretical analysis (cf. Sections 2.3 and 3). This further highlights the overall better performance of GC as suggested in Section 4.2.2 and confirmed in Table 1. We do not have a clear reason why LSUN is an exception. Note however that GC and OT are nearly in the same confidence interval of the FID on this dataset, while GC is significantly superior to batch-OT on CIFAR-10 and CelebA.

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2. As a weakness, the reported empirical results differ significantly from the original baseline iCT. One problem is surely due to the lack of an open source official implementation of iCT. I would suggest the authors to improve their iCT baseline, and to include more experiments based on ECT, as the experiments are relatively fast and can be done with small batch size. [emphasis ours]

The difference in FID values stems from the following differences: the original iCT baseline uses 400k iterations with a batch size of 1024, while we use 100k iterations with a batch size of 512. Still, the comparison of all models in our paper is fair: they are trained and evaluated in the same setting, yielding sound conclusions. Moreover, in the compute-efficient ECT setting, we are still superior to independent coupling. As recommended, we provide new results in the ECT setting on a larger-resolution ImageNet and FFQH (new dataset) that we will include in Section 5.3:

ECT Results (FID)

Overall, GC is superior to IC on all datasets for iCT (4/4), and on 5/6 settings for ECT. Note that in the ECT setting, we could run the experiments with the same training time than in the original paper, and thus report similar FID values. This validates the benefits of using GC in practice.

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3. I wonder if the authors have tried annealing the parameter from $0$ to $1$ during training. Intuitively, I would expect that at early training iterations, the model would benefit from using mostly the independent coupling, as the learned solution is probably still quite off, and then as training progresses and the sample quality improves, the model can rely more and more on the generator. [emphasis ours]

Indeed, this is an interesting question. We did experiment with this type of annealing which resulted in lower performance than with a fixed $\mu$. One explanation is that it does not interact well with the scheduling on the number of timesteps that is periodically increased during training: the generator is not optimal on newly introduced timesteps.

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4. I would suggest adding "Minimizing Trajectory Curvature of ODE-based Generative Models" from Sangyun Lee, Beomsu Kim, and Jong Chul Ye, to the discussion of coupling methods in section 3.

We thank the reviewer for this interesting reference which we will include in our discussion. In this work, the authors propose to learn an encoder from data to noise, and use this encoder as a way to construct a coupling when training a flow model. In comparison to our method, their algorithm is based on a joint training of the velocity field and of the noise encoder which then requires an additional network.

We would like to thank the reviewer for their detailed and positive assessment. We address the raised questions/weaknesses below.

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1. The experiments cover moderate-scale. Could the authors add an experiment on $256\times256$? [emphasis ours]

We cannot conduct experiments on larger resolutions from scratch for computational reasons. Moreover, the only pre-trained diffusion models available in the compute-efficient ECT setting are on ImageNet $512 \times 512$, which is also too large given our resources.

Still, we were able to obtain new results on a larger-resolution ImageNet ($64\times 64$ conditional) and a new dataset (FFHQ $64\times 64$) in the ECT setting that we will include in Section 5.3:

ECT Results (FID) on ImageNet-64 and FFHQ-64

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2. The concept partly assumes an “ideal” or well-trained generator for the GC approach. The paper addresses this with “joint learning,” but theoretically that is more complicated, and some of the formal results rely on the generator’s accuracy. What do you think about this?

Indeed, our theoretical assumption requires an ideal generator to construct GC. Our joint learning approach allows to approximate this ideal generator and, when the generator is close enough to an ideal one, joint learning succeeds. Interestingly, we show that when the generator is not trained enough on IC, then GC does not work anymore (see Section C.1 and Figure 7 in Appendix). This proves the effectiveness of the joint learning approach: the approximation is good enough to construct GC trajectories.

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3. Could the approach generalize easily to other cost functions in the coupling—e.g., $L_1$ or robust divergences? Does the preference for quadratic cost (in the “transport cost” sense) strictly matter for performance in your experiments? [emphasis ours]

GC is agnostic to all distance functions, whether the one used in consistency model training (Eq. 4/6/15) or the transport cost used in the analysis in Section 4.2.2. Regarding the transport cost, the algorithm is not built to minimize this distance directly, as opposed to batch-OT. We did not study this further but intuitively expect GC to also minimize transport cost with regards to other cost functions.

We would be happy to provide further clarification, should we have misinterpreted to the reviewer's question.