reviews: 
Comments: This paper tackles the problem of  instability in continuous consistency models and presents many contributions. 

The first novelty is the simplification of model parametrization in the EDM/CM frameworks. It is shown that rescaling the mean ($\alpha_t$ ) and noise ($\sigma_t$) schedule of conditional probability flows with the norm $||(\alpha_t, \sigma_t)||$, produces a second conditional flow, such that the EDM model parametrization and loss in the first are identical to those in the second. Furthermore, the ODE sampling procedure remains unchanged. However, the connection between the new flow schedule parameters and the  EDM scaling parameters is greatly simplified after normalization, facilitating further theoretical analysis, while maintaining the benefits of the EDM model definition (unit variance of input, target and minimizing the scaling of the output). The formulation with $\alpha_t=cos(t)$ and  $\sigma_t=sin(t)$ is named *TrigFlow*.

Then the training objective is studied and probed for causes of instability. Several factors are detected such as: an inappropriate $c_{noise}(t)$ which is fixed by being set to $t$; inappropriate Fourier scales in the time positional embeddings which are then reduced; the usage  'AdaGN layer', which is then replaced with 'Adaptive Double Normalization'; highly varying target norm alleviated by target normalization; inappropriate weighting mitigated by adaptive weighting and unstable terms solved by slowly introducing such terms into the loss with respect to the number of parameter updates. 

In addition, methods for scaling such models to large sizes and datasets are proposed, namely JVP Rearrangement and JVP of Flash Attention.

Finally, the proposed models are compared against the state of the art methods, and results show that continuous consistency models are excellent generators, that only require 1 or 2 sampling steps to generate new quality data from learned distributions with continuous support.
Strengths: The contributions of this paper are novel, clear, significant and positioned correctly.

Trigflow simplifies the theoretical analysis of flows by showing that  the mean ($\alpha_t$ ) and noise ($\sigma_t$) schedule in the definition of conditional flows can be normalized while preserving the model/loss formulation and the integrator-generated paths, while simplifying the connection between the scaling parameters of the model and those of the conditional flow. **(novelty, significance)**

Several components of continuous consistency models are studied and probed for potential causes of instability. They include: an inappropriate $c_{noise}(t)$ which is fixed by being set to $t$; inappropriate Fourier scales in the time positional embeddings which are then reduced; the usage  'AdaGN layer', which is then replaced with 'Adaptive Double Normalization'; highly varying target norm alleviated by target normalization; inappropriate weighting mitigated by adaptive weighting and unstable terms solved by slowly introducing such terms into the loss with respect to the number of parameter updates.  **(novelty, significance)**

Strategies for scaling such models to large sizes and datasets are proposed, namely JVP Rearrangement and JVP of Flash Attention. **(quality, significance)**

The proposed method shows competitiveness with the state of the art models, while only using 1 or 2 steps for generation, and outperforms all other tested methods that use 1 to 2 generation steps **(significance, quality)**
Weaknesses: **While the contributions of the paper are numerous, the paper could be strengthened even further by:**

1) Comparing the proposed model with more recent flow models such as [1] and [2] and adding the results for rectified flows with 2 generation steps.

2) Placing the number of parameters and the number of parameter updates (or training time on identical hardware) for each model in Table 1 is very important as using an equal umber of parameters/compute is essential for ensuring a fair comparison.

**Some smaller issues and suggestions:**

a) The paper would be improved if an intuitive explanation is given for the loss in Equation 2. Also it could save readers some time if it is mentioned that the loss is derived in Song et al 2023, *Remark 10*. 

b) The paper would be enriched by adding some generated images with one step.

c) Shouldn't $c_{skip}$ and $c_{out}$ be $\sigma_t$ and  $ \alpha_t \sigma_d$, that is for $\sigma_t=t$ and  $\alpha_t=1$: $c_{skip}=t$  and $c_{out}=\sigma_d$ in line 201/202?

d) In Equation (20) Appendix, shouldn't it be $\hat{D}$ for $D_{\theta}(x_t)=\hat{D}_{\theta}(\hat{x}_t(x_t))$?

e) The paragraph from line 924 to 938 (appendix) needs additional elaboration regarding the implications of having $||(\alpha_t, \sigma_t)||=1$ with respect to the invariance of the geometric set connecting $x_0$ and $z$.



**Typos:**

i) In line 126, a 2 is squared instead of the norm.

ii) in line 122, $z_t$ does not depend on time.


**References**


[1]  Tong et al. 2024. Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport

[2]  Kornilov et al 2024. Optimal Flow Matching: Learning Straight Trajectories in Just One Step
Recommendation: 10
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 3