Strengths
1. The paper is easy to follow, the presentation from concepts to methods is concise, the motivation of the proposed method on overcoming shortage of flow model is also clear and straightforward.
2. Incorporating the flow prior into MAP to solve inverse problems is interesting and worth exploring.
3. The discussion on feasibility of assumptions made in the paper, as well as hyperparameter effects are very informative to the readers, and are valuable to the research in the community.
Weaknesses
1. Theorem 1 is trivial in terms of $N \rightarrow 0$, and the existence of constant $c(N)$. The approximation gap converges to 0 as N goes to infinity does not provide insightful guidance on the practical implementation since $N \rightarrow 0$ is infeasible (decreases efficiency) in practice. Also, the implicit representation of approximation error constant $c(N)$ does not make sense because it gives no information about how the error could be scaled with $N$, the authors are expected to provide explicit expression of $c( \cdot )$ as a function of $N$.
2. Obtaining measurements $u_t$ in the ‘corrupted trajectory’ and generating auxiliary path $s_t$ do not make too much sense to me, especially when $x_t$ is very noisy and the forward operator $A(\cdot)$ is challenging. Suppose $A$ is very ill-posed (e.g. compressed sensing with low sampling rate), then the $y_t$ in the auxiliary path is not good itself, nevertheless the measurement $u_t = A(x_t)$ obtained in the corrupted trajectory especially when $x_t$ has high noise level. The assumption on exact compliance of trajectory (between corrupted trajectory and auxiliary path) is not theoretically guaranteed as stated by the authors (line 170-171), and the strong empirical results can not support the existence of the assumption made. The good empirical results could only be valid in less challenging inverse problems, or those inverse problems that are suitable for generative models to solve, e.g. inpainting, super-resolution.
3. In the compressed sensing experiment, the sampling rate $\nu$ is relatively large (0.25, 0.5). The authors are expected to use smaller sampling rates (e.g. 0.05, 0.1), and at the same time compare it with classical recovery algorithms, developed from the seminal work by Donoho, et al [1,2] (no neural networks involved), as well as compressed sensing with other generative models (VAE, GAN, Diffusion). The current results and comparison are not convincing.
[1] Donoho, David L. "Compressed sensing." IEEE Transactions on information theory 52.4 (2006): 1289-1306.
[2] Lustig, Michael, et al. "Compressed sensing MRI." IEEE signal processing magazine 25.2 (2008): 72-82.
4. The $\lambda_t$ used in Proposition 1 as SNR can also be found in previous work [1], the authors are expected to provide some comparison or discussion with [1] although it is based on Diffusion, it uses MAP structure and solve optimization problem during the sampling process.
[1] Mardani, Morteza, et al. "A variational perspective on solving inverse problems with diffusion models." arXiv preprint arXiv:2305.04391 (2023).
5. The number of iterations $K$ affects the performance a lot on different tasks, the authors are expected to provide more explanation on the choice of $K$. Also, how are the $K$ and $\lambda$ picked in the experiments? Given that the testing performances are sensitive to these hyperparameters.
6. The baseline methods compared are limited, only 3 flow based models are picked, generative models have been widely used for image restoration tasks, the authors are expected to provide at least several other representative generative models for comparison.