1. I find that separating Theorem 3.3 into parts A and B is tangential to the story and overly confusing. In reality, we do not have full control over our correction network and its Lipschitz constant. Therefore, we can never determine the best scheduling. This section seems like its being theoretical for its own sake! It might be clearer to simply present Lemma A.2 of the appendix in its most general form:
$$W_2(p^b_t, p^f_t) \le W_2(p^b_{t_0}, p^f_{t_0}) \cdot e^{L(t-t_0)}$$
and say that improving the Wasserstein distance on the RHS for $t_0$ can effectively bound the Wasserstein distance on the LHS, especially for $t$ that is sufficiently close to $t_0$. I don't think A, B, and examples 3.4 and 3.5 are particularly insightful when it is not directly factored into the decisions made in the experiments. The results in A and B can still be included, but in the appendix.
2. The parallel sampling section seems slightly oversold! To my understanding, while both forward passes can be done in parallel, it cannot be done in one batch because the forward call is on different methods. Could you please provide a time comparison between parallel and serial sampling on one experiment with the hardware that you have?
3. The statement of Lemma 3.6 seems to spill over to the rest of the main text and I generally do not agree with the base assumption that $p_t^f = p^b_{t, \sigma}$ which is the main driver for Lemma 3.6. Please let me know if I am misunderstanding this!
4. I don't find the comparison between this method and Dai and Wipf [B] appropriate! [B] trains a VAE on VAE to fix problems associated with the dimensionality mismatch between the data manifold and the manifold induced by the (first) VAE. That is not a concern in flow-matching and diffusion models as these models are known not to suffer from the manifold mismatch difficulties as much.
5. Although FIDs are still being widely used for evaluation, there have been clear flaws associated with them and the simplistic Inception network [C]. Please use DinoV2 Frechet Distances for the comparisons from [C], in addition to the widely used FID metric.
6. Please also provide evaluations "matching" the same NFEs in the corresponding non-corrected models.
### Minor points
1. I personally do not agree with the notation abuse of rewriting the conditional probability flow $p_t(x | z)$ as the marginal probability flow $p_t(x)$; it is highly confusing in my opinion.
2. Rather than introducing the new probability flows $\nu_t$ and $\mu_t$, in theorem 3.3, please consider using the same $p^b_t$ and $p^f_t$ for reduced notation overhead, and then restate the theorem in full formality for the appendix.
3. (nitpick) In Eq. (8), $t$ should be a sub-index of $u$.
5. Although FIDs are still being widely used for evaluation, there have been clear flaws associated with them and the simplistic Inception network [C]. Please use DinoV2 Frechet Distances for the comparisons from [C], in addition to the widely used FID metric.