Flow Priors for Linear Inverse Problems via  Iterative Corrupted Trajectory Matching

Authors: yasi  zhang
peiyu  yu
yaxuan  zhu
yingshan  chang
feng  gao
ying nian wu
oscar  leong
Venue: NeurIPS_cc_2024_Conference
Published: 2024-04-30 00:40:52

Abstract: Generative models based on flow matching have attracted significant attention for their simplicity and superior performance in high-resolution image synthesis. By leveraging the instantaneous change-of-variables formula, one can directly compute image likelihoods from a learned flow, making them enticing candidates as priors for downstream tasks such as inverse problems. In particular, a natural approach would be to incorporate such image probabilities in a maximum-a-posteriori (MAP) estimation problem. A major obstacle, however, lies in the slow computation of the log-likelihood, as it requires backpropagating through an ODE solver, which can be prohibitively slow for high-dimensional problems. In this work, we propose an iterative algorithm to approximate the MAP estimator efficiently to solve a variety of linear inverse problems. Our algorithm is mathematically justified by the observation that the MAP objective can be approximated by a sum of $N$ ``local MAP'' objectives, where $N$ is the number of function evaluations. By leveraging Tweedie's formula, we show that we can perform gradient steps to sequentially optimize these objectives. We validate our approach for various linear inverse problems, such as super-resolution, deblurring, inpainting, and compressed sensing, and demonstrate that we can outperform other methods based on flow matching. Code is available at \url{https://github.com/YasminZhang/ICTM}.

Main paper: 
 introduction: 
 Linear inverse problems are ubiquitous across many imaging domains, pervading areas such as astronomy [41, 23] , medical imaging [38, 49] , and seismology [35, 39] . In these problems the goal is to reconstruct an unknown image x * ∈ R n from observed measurements y ∈ R m of the form:EQUATIONwhere A : R n → R m with m ≤ n is a linear operator that degrades the clean image x * , and the additive noise is drawn from a known distribution. In this work, we assume the noise follows N (0, σ 2 y I). Due to the under-constrained nature of such problems, they are typically ill-posed, i.e., there are an infinite number of undesirable images that fit to the observed measurements. Hence, one requires further structural information about the underlying images, which constitutes our prior.With the advent of large generative models [27, 17, 48, 40, 8, 58] , there has been a surge of interest in exploiting generative models as priors to solve inverse problems. Given a pretrained generator to sample from a distribution or grant access to image probabilities, one can solve a variety of inverse problems in a task-or forward model-agnostic fashion, without the need for large-scale supervision [36] . This has been successfully done for a variety of models, including implicit generators such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) [4, 34] , invertible generators such as Normalizing Flows [1, 54] , and more recently Diffusion models [10, 43, 59] .A recent paradigm in generative modeling [48, 55, 25, 57] , based on the concept of flow matching [29, 28] , has made significant strides in scaling ODE-based generators to high-resolution images.Flow matching models map a simple base distribution, such as a Gaussian, to a complex, highdimensional data distribution by defining a flow field that represents the transformation between these distributions. These generative models have demonstrated scalability to high dimensions, forming the backbone of several state-of-the-art generative models [30, 13, 56] . Moreover, flow matching models follow straighter and more direct probability paths compared to diffusion models, allowing for more efficient and faster sampling [28, 29, 13] . Additionally, due to their invertibility, flow matching models provide direct access to image likelihoods through the instantaneous change-of-variables formula [9, 18] . Given these advantages and the relatively recent application of these models to inverse problems [37, 2] , we investigate their use as image priors in this work.Leveraging knowledge about the corruption process p(y|x) and a natural image prior p(x), the Bayesian approach suggests analyzing the image reconstruction posterior p(x|y) ∝ p(y|x)p(x) to solve the inverse problem . A proven and effective method based on this approach is maximum-aposteriori (MAP) estimation [6, 19] , which maximizes the posterior to identify the image most likely to match the observed measurements: argmin MAP estimation provides a single, most probable point estimate of the posterior distribution, making it simple and interpretable. This deterministic approach ensures consistency and reproducibility, which are essential in applications requiring reliable outcomes, particularly in compressed sensing tasks such as Computed Tomography (CT) [7] and Magnetic Resonance Imaging (MRI) [52] . While posterior sampling methods can offer diverse reconstructions to quantify uncertainty, they can be prohibitively slow in high-dimensions [5] . Hence, in this work, we propose to integrate flow priors to solve linear inverse problems by MAP estimation.A significant challenge in employing flow priors for MAP estimation lies in the slow computation of the image probabilities, as it requires backpropagating through an ODE solver [47, 16, 15] . In this work, we show how one can address this challenge via Iterative Corrupted Trajectory Matching (ICTM), a novel algorithm to approximate the MAP solution in a computaionally efficient manner. In particular, we show how one can approximately find an MAP solution by sequentially optimizing a novel simpler, auxillary objective that approximates the true MAP objective in the limit of infinite function evaluations. For finite evaluations, we demonstrate that this approximation is sufficient to optimize by showcasing strong empirical performance for flow priors across a variety of linear inverse problems. We summarize our contributions as follows:1. We propose ICTM, an algorithm to approximate the MAP solution to a variety of linear inverse problems using a flow prior. This algorithm optimizes an auxillary objective that partitions the flow model's trajectory into N "local MAP" objectives, where N is the number of function evaluations (NFEs). By leveraging Tweedie's formula, we show that we can perform gradient steps to sequentially optimize these objectives. 2. Theoretically, we demonstrate that the auxillary objective converges to the true MAP objective as the NFEs goes to infinity. We validate the correctness of our algorithm in finding the MAP solution on a denoising problem. 3. We demonstrate the utility of ICTM on a wide variety of linear inverse problems on both natural and scientific image datasets, with problems including denoising, inpainting, superresolution, deblurring, and compressed sensing. Extensive results show that ICTM is both computationaly efficient and obtains high-quality reconstructions, outperforming other reconstruction algorithms based on flow priors. 
background: 
 Notation We follow the convention for flow-based models, where Gaussian noise is sampled at timestep 0, and the clean image corresponds to timestep 1. Note that this is the opposite of diffusion models. For t ∈ [0, 1], we denote x t (x 0 ) as the point at time t whose initial condition is x 0 . In this work, we use x and x 1 interchangeably, i.e., x 1 (x 0 ) = x(x 0 ).We consider generative models that map samples x 0 from a noise distribution p(x 0 ), e.g., Gaussian, to samples x 1 of a data distribution p(x 1 ) using an ordinary differential equation (ODE):EQUATIONwhere the velocity field v is a θ-parameterized neural network, e.g., using a UNet [28, 29, 42] or Transformer [13, 51] architecture. Generative models based on flow matching [28, 29] can be seen as a simulation-free approach to learning the velocity field. This approach involves pre-determining paths that the ODE should follow by specifying the interpolation curve x t , rather than relying on the MLE algorithm to implicitly discover them [9] . To construct such a path, which is not necessarily Markovian, one can define a differentiable nonlinear interpolation between x 0 and x 1 :EQUATIONwhere both α t and β t are differentiable functions with respect to t satisfying α 0 = 0, β 0 = 1, and α 1 = 1, β 1 = 0. This ensures that x t is transported from a standard Gaussian distribution to the natural image manifold from time 0 to time 1. In contrast, the diffusion process [48, 45, 20] induces a non-differentiable trajectory due to the diffusion term in the SDE formulation.The idea behind flow matching is to utilize the power of deep neural networks to efficiently predict the velocity field at each timestep. To achieve this, we can train the neural network by minimizing an L 2 loss between the sampled velocity and the one predicted by the neural network:EQUATION)We denote the optimal (not necessarily unique) solution to arg min θ L(θ) as θ. The optimal velocity field v θ can be derived in closed form and is the expected velocity at state x t :EQUATIONFor convenience, in the following text, we use v θ to refer to the optimal v θ . In the rest of the paper, we assume that the flow v θ and its parameters are pretrained on a dataset of interest and fixed. We are then interested in leveraging its utility as a prior to solve inverse problems.Denote the probability of x t in Eq. (3) as p(x t ) dependent on time. Assuming that v θ is uniformly Lipschitz continuous in x t and continuous in t, the change in log probability also follows a differential equation [9, 18] :EQUATIONOne can additionally obtain the likelihood of the trajectory via integrating Eq. ( 7) across timeEQUATION 
method: 
 In this work, we aim to solve the MAP estimation problem in Eq. ( 2) where p(x) is given by a pretrained flow prior. We first discuss in Section 3.1 how the MAP problem could, in principle, be solved via a latent-space optimization problem. As we will see, this problem is challenging to solve computationally due to the need to backpropagate through an ODE solver. To overcome this, we show in Section 3.2 that the ideal MAP problem can be approximated by a weighted sum of "local MAP" optimization problems, which operates by partitioning the flow's trajectory to a reconstructed solution. We then introduce our ICTM algorithm to sequentially optimize this auxiliary objective. Finally, in Section 3.3, we experimentally validate that our algorithm finds a solution that is faithful to the MAP estimate in a simplified setting where the globally optimal MAP solution is known.Given a pretrained flow prior, one can compute the log-likelihood of x generated from an initial noise sample x 0 via Eq. (8) . Hence, to find the MAP estimate, one could equivalently optimize the initial point of the trajectory x 0 and return x 1 (x 0 ) where x 0 is found by solvingEQUATIONwhere x t := x t (x 0 ) denotes the intermediate state x t generated from x 0 . Intuitively, this loss encourages finding an initial point x 0 such that the reconstruction x 1 := x 1 (x 0 ) fits the observed measurements, but is also likely to be generated by the flow.In practice, x 1 and the prior term can be approximated by an ODE solver. The trajectory of x t = x 0 + t 0 v θ (x t , t)dt can be approximated by an ODE sampler, i.e. ODESolve(x 0 , 0, t, v θ ), where x 0 is the initial point, and the second and third arguments represent the starting time and the ending time, respectively. For example, with an Euler sampler, we iterate over x t+∆t = x t + v θ (x t , t)∆t where ∆t = 1/N and N is the predetermined NFEs. After acquiring the optimal x0 by optimizing the Eq. ( 9), we obtain the MAP solution x 1 by using ODESolve(x 0 , 0, 1, v θ ) again.The global flow-based MAP objective Eq. ( 9) is tractable for low-dimensional problems. The challenge for high-dimensional problems, however, is that optimizing Eq. ( 9) is simulation-based, and thus each update iteration requires full forward and backward propagation through an ODE solver, resulting in issues regarding memory inefficiency and time, making it hard to optimize [9, 15, 16, 47] .As a way to address this, we prove a result in Theorem 1 that shows that the MAP objective can be approximated by a weighted sum of N local posterior objectives. These objectives are "local" in the sense that they mainly depend on likelihoods and probabilities of intermediate trajectories x t and x t + v θ (x t , t)∆t for t = 0, ∆t, . . . , N ∆t where ∆t := 1/N . Given an initial noise input x 0 , each local posterior objective depends on a non-Markovian auxiliary path y t = α t y + β t A(x 0 ) by connecting the points between y and Ax 0 . We prove this result for straight paths α t = t and β t = 1 -t for simplicity, but other interpolation paths can be used. The proof is in Section A.2. Theorem 1. For N ≥ 1, set γ i := ( 12 ) N -i+1 and ∆t = 1/N . Suppose y = A(x * ) + ϵ where x * = x 1 (x 0 ) with x 0 being the solution to Eq. (9), ϵ ∼ N (0, σ2 y I), and x t exactly follows the straight pathx t = tx + (1 -t)x 0 for any timestep t ∈ [0, 1]. Suppose the velocity field v θ : R n × R → R n satisfies sup z∈R n ,s∈[0,1] |tr ∂ ∂x v θ (z, s)| ≤ C 1 for some universal constant C 1 .Then, there exists a constant c(N ) 2 that does not depend on x 0 such thatlim N →∞ log p(x(x 0 )|y) - N i=1 γ i Ĵi -c(N ) = 0,where Ĵi = log p(x (i-1)∆t ) -tr ∂v θ (x (i-1)∆t ,(i-1)∆t) ∂x ∆t + log p(y i∆t |x i∆t ).This result shows that the true MAP objective evaluated at the optimal solution can be approximated by a weighted sum of objectives that depend locally at a time t for the trajectory {x t : t ∈ [0, 1]}. The intuition regarding Ĵi arises from the fact that Ĵi ≈ J i , where J i is the local posterior distributionJ i = log p(y i∆t |x i∆t (x (i-1)∆t )) + log p(x i∆t ).Optimizing each of these local posterior distributions in a sequential fashion captures the fact that we would like each intermediate point in our trajectory x i∆t to be likely and fit to our measurements, ideally resulting in a final reconstruction x 1 that satisfies this as well. The benefit of Ĵi , as we will show in the sequel, is that it is efficient to optimize.We assume that the trajectory {x t } t exactly follows the predefined interpolation path {α t x + β t x 0 } t . In Section B of the appendix, we analyze this assumption and show that we can bound the deviation from the predefined interpolation path to the learned path via a path compliance measure. Moreover, we impose a regularity assumption on the velocity field v θ , effectively requiring a uniform bound on the spectrum of the Jacobian of v θ . This can be easily satisfied with neural networks using Lipschitz continuous and differentiable activation functions.As we see in Theorem 1, one can approximate the true MAP objective via a sum of local objectives of the formEQUATION)At first glance, Ĵi still appears challenging to optimize, but there are additional insights we can exploit for computation. We discuss each term in Ĵi below.The intuition behind ICTM is that we aim to match a corrupted trajectory {u t } t with an auxiliary path {y t } t specified by an interpolation between our measurements y and A(x 0 ) for each timestep t, defined by y t := α t y + β t A(x 0 ). The corrupted trajectory u t := A(x t ) follows the corrupted flow ODE du t = A(v θ (x t , t))dt. To optimize the above "local MAP" objectives, we must understand the distribution of p(y t |x t ). Generally speaking, this distribution is intractable. However, by assuming exact compliance of the trajectory generated by flow to the predefined interpolation path (as done in Theorem 1), we can show that y t |x t ∼ N (u t , α 2 t σ 2 y ). This is proven in Lemma 3 in the appendix. While exact compliance of the trajectory may not hold for learned flow matching models, we show empirically that making this assumption leads to strong performance in practice. We further analyze this notion of compliance in Section B of the appendix.The approximation in Eq. ( 10) addresses one of the main concerns of MAP in that the intensive integral computation is circumvented with a simpler Riemannian sum. This approximation holds for small time increments ∆t:t+∆t t tr ∂ ∂x v θ (x s , s) ds ≈ tr ∂ ∂x v θ (x t , t) ∆t.Note that one can additionally improve the efficiency of this term by employing a Hutchinson-Skilling estimate [44, 21] for the trace of the Jacobian matrix. However, at first glance, it appears we have simply shifted the problem to the computation of the prior at timestep (i -1)∆t. Fortunately, it is possible to derive a formula for the gradient of log p(x t ) for all timesteps t ∈ [0, 1] using Tweedie's formula [12] . This allows us to optimize each objective Ĵi using gradient-based optimizers. The following result gives a precise characterization of ∇ xt log p(x t ), proven in Section A.1. Proposition 1. Let λ t = α t /β t denote the signal-to-noise ratio. The relationship between the score function ∇ xt log p(x t ) and the velocity field v θ (x t , t) is given by:EQUATIONIn summary, we have derived an efficient approximation to the MAP objective. For our algorithm, we iteratively optimize each term Ĵt sequentially for each t = 0, ∆t, . . . , N ∆t, fitting our current iterate x t to induce an increment x t+∆t such that A(x t+∆t ) fits to our auxiliary corrupted path y t+∆t while being likely under our local prior. We call this approach Iterative Corrupted Trajectory Matching (ICTM). Our algorithm is summarized in Algo. 1. In lines 7 and 12, instead of directly optimizing the local data likelihood, we choose λ as a new hyper-parameter to tune. We find a constant λ works well in practice.We experimentally validate that the reconstruction found via ICTM is close to the optimal MAP solution in a simplified denoising problem where the MAP solution can be obtained in closed-form. 1: Initialize ϵ ∼ N (0, I), x 0 ← ϵ, t ← 0, ∆t ← 1/N 2:Generate an auxiliary path y s = α s y + β s (Ax 0 ) for s ∈ (0, 1) 3: while t < 1 do 4:x t+∆t ← x t + v θ (x t , t)∆t 5:if t = 0 then 6:for k = 1, • • • K do 7: x t ← x t -η∇ xt λ∥A(x t+∆t (x t )) -y t+∆t ∥ 2 + 1 2 ∥x t ∥ 2 + tr ∂v θ (xt,t) ∂x ∆t 8:end for 9:else 10:for k = 1, • • • K do 11:# use Eq. ( 11) to obtain the gradient of log p(x t )12:x t ← x t -η∇ xt λ∥A(x t+∆t (x t )) -y t+∆t ∥ 2 -log p(x t ) + tr ∂v θ (xt,t) ∂x ∆t 13:end for 14:end if 15:x t+∆t ← x t + v θ (x t , t)∆t Specifically, we fit a Gaussian distribution N (µ, Σ) using 1,000 samples from the FFHQ dataset. Consider a denoising problem y = x + ϵ where x ∼ N (µ, Σ) and ϵ ∼ N (0, σ 2 y I). In this case, the analytical solution to the MAP estimation problem (Eq. ( 2)) is x * = (Σ -1 +σ -2 y I) -1 (Σ -1 µ+σ -2 y y). We set σ y = 0.1. Then, we train a flow-based model on 10,000 samples from the true Gaussian distribution and showcase the deviation of our reconstruction found via ICTM to the closed-form MAP solution x * in Fig. 1 . We see that ICTM can obtain a faithful estimate of the MAP solution across many samples. 
experiments: 
 In our experimental setting, we use optimal transport interpolation coefficients, i.e. α t = t and β t = 1 -t. We test our algorithm on both natural and medical imaging datasets. For natural images, we utilize the pretrained checkpoint from the official Rectified Flow repository3 and evaluate our approach on the CelebA-HQ dataset [31, 24] . We address four common linear inverse problems: super-resolution, inpainting with a random mask, Gaussian deblurring, and inpainting with a box mask.For the medical application, we train a flow-based model from scratch on the Human Connectome Project (HCP) dataset [50] and test our algorithm specifically for compressed sensing at different compression rates. Our algorithm focuses on the reconstruction faithfulness of generated images, therefore employing PSNR and SSIM [53] as evaluation metrics.Baselines We compare our method with five baselines. 1) OT-ODE [37] . To our knowledge, this is the only baseline that applies flow-based models to inverse problems. They incorporate a prior gradient correction at each sampling step based on conditional Optimal Transport (OT) paths. For a fair comparison, we follow their implementation of Algorithm 10) in our algorithm. In the experiments with natural images, in addition to the flow-based baselines, we have included two representative diffusion-based baselines: 4) RED-Diff [33] , a variational Bayes-based method; and 5) ΠGDM [46] , an advanced MCMC-based method. We also note one concurrent work, D-Flow [2] , which formulates the MAP as a constrained optimization problem in their Eq. 9. As documented in their Sec. 3.4, it takes 5-10 minutes to recover each image. This is because each of its optimization step requires backpropagation through an ODE solver to compute the full log-likelihood. In contrast, our method is significantly faster (approximately 1.6 minutes per image) due to our principled local MAP approximation, as demonstrated in Appendix D.Experimental setup We evaluate our algorithm using 100 images from the CelebA-HQ validation set with a resolution of 256×256, normalizing all images to the [0, 1] range for quantitative analysis. We present the quantitative and qualitative results of all the methods in Tab. 1 and Fig. 2 , respectively. In Tab. 1, our method surpasses all other baselines across all tasks. For more challenging tasks such as Gaussian deblurring and box inpainting, our method significantly outperforms others in terms of SSIM. Based on the MAP framework, as shown in Fig. 2 , our method prefers more faithful and artifact-free reconstructions, whereas others trade off for perceptual quality. We note that there is an unavoidable tradeoff between perceptual quality and restoration faithfulness [3] . Overall, our method presents a higher degree of refinement. The comparison between ours and ours (w/o prior) indicates the effectiveness of the local prior term in enhancing the accuracy of the reconstructions, as evidenced by the increases in both PSNR and SSIM.HCP T2w dataset We utilize images from the publicly available Human Connectome Project (HCP) [50] T2-weighted (T2w) images dataset for the task of compressed sensing, which contains brain images from 47 patients. The HCP dataset includes cross-sectional images of the brain taken at different levels and angles.Compressed sensing We train a flow-based model from scratch on 10,000 randomly sampled images, utilizing the ncsnpp architecture [48] with minor adaptations for grayscale images. We employ compression rates ν ∈ {1/2, 1/4, 1/10}, meaning m = νn. The measurement operator is given by a subsampled Fourier matrix, whose sign patterns are randomly selected. We evaluate our reconstruction algorithm's performance on 200 randomly sampled test images. For super-resolution and the other three tasks, K = 1 is sufficient to achieve the best performance with the optimal step size η and guidance weight λ. However, for compressed sensing, it is necessary to increase K to obtain the best performance. We hypothesize that this is due to the increased complexity of the compressed sensing operator, which requires more iteration steps to ensure the correct optimization direction.We present the quantitative and qualitative results of compressed sensing in Tab. 2 and Fig. 3 , respectively. In addition to flow-based methods, we include results for two classical recovery algorithms, Wavelet [11, 32] and TV [22] priors. As shown in Tab. 2, our method outperforms the classical recovery algorithms and other flow-based baselines across varying compression rates ν, demonstrating our method's capability to handle challenging scenarios and the advantages of utilizing modern generative models as priors. In Fig. 3 , our method produces reconstructions that are more faithful to the original images, with fewer artifacts, leading to higher accuracy and clearer details.We use the Adam optimizer [26] for our optimization steps due to its effectiveness in neural network computations. For all tasks, we utilize N = 100 steps.Step size η and Guidance weight λ The use of the Adam optimizer ensures that the choice of hyperparameters, particularly the step size η and the guidance weight λ, remains consistent across various tasks, as illustrated in Fig. 4 . Specifically, a step size of η = 10 -2 is optimal for Inpainting (random), Inpainting (box), and Super-resolution in terms of SSIM. For PSNR, Gaussian deblurring also achieves optimal performance at η = 10 -2 . Consequently, we employ η = 10 -2 for all tasks.Based on the results shown in the right two subfigures of Fig. 4 , we select λ = 10 3 for Gaussian deblurring and λ = 10 4 for the other tasks. This consistency extends to the compressed sensing experiments, where we set λ = 10 3 and η = 10 -2 for all experiments involving medical images.Iteration number K We present ablation results of the iteration number K on different tasks in Fig. 5 . We focus on the behavior of K in super-resolution and compressed sensing, as it performs similarly to super-resolution in the other three tasks. With the optimal choice of η and λ in super-resolution, i.e., η = 10 -2 and λ = 10 3 , K = 1 provides superior performance on CelebA-HQ. A decreased step size, e.g., η = 10 -3 , can help performance as K increases, but it fails to exceed the performance achieved with the optimal parameters at K = 1. However, for compressed sensing, it is necessary to increase K to achieve the best performance. Consequently, we set K = 10 for all compressed sensing experiments. We hypothesize that the complexity of the compressed sensing operator directly determines the number of iterations required for optimal performance. 
conclusion: 
 In this work, we have introduced a novel iterative algorithm to incorporate flow priors to solve linear inverse problems. By addressing the computational challenges associated with the slow log-likelihood calculations inherent in flow matching models, our approach leverages the decomposition of the MAP objective into multiple "local MAP" objectives. This decomposition, combined with the application of Tweedie's formula, enables effective sequential optimization through gradient steps.Our method has been rigorously validated on both natural and scientific images across various linear inverse problems, including super-resolution, deblurring, inpainting, and compressed sensing. The empirical results indicate that our algorithm consistently outperforms existing techniques based on flow matching, highlighting its potential as a powerful tool for high-resolution image synthesis and related downstream tasks.Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: We have provided sufficient implementation details and links for original repositories.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
limitations and future work: 
 While our algorithm has demonstrated promising results, there are certain limitations that suggest avenues for future research. First, our theoretical framework, built on optimal transport interpolation paths, is currently limited and cannot be applied to solve the general interpolation between Gaussian and data distributions. Additionally, in order to broaden the applicability of flow priors for inverse problems, it is important to generalize our approach to handle nonlinear forward models. Moreover, the algorithm currently lacks the capability to quantify the uncertainty of the generated images, an aspect crucial for many scientific applications. It would be interesting to consider approaches to post-process our solutions to understand the uncertainty inherent in our reconstruction. These limitations highlight important directions for future work to enhance the robustness and applicability of our method.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: They are provided in Section Experiments and Appendix.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes]Justification: Standard deviations are provided in Tables.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: They are provided in Appendix.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. • Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: Before we dive into the proof, we provide the following three lemmas. Lemma 1. Consider a vector-valued function f : [0, 1] → R n . Then for any t ∈ [0, 1], we have thatt 0 f (s)ds 2 ≤ t 0 ∥f (s)∥ 2 ds.Proof. For each s ∈ [0, 1], let f i (s) ∈ R denote the i-th component of f (s). Recall Jensen's inequality: for any convex function g : R → R and integrable function h : [0, 1] → R, we haveg b a h(t)dt ≤ b a g(h(t))dt.Using convexity of the function t → t 2 and applying Jensen's inequality, we see thatt 0 f (s)ds 2 = n i=1 t 0 f i (s)ds 2 ≤ n i=1 t 0 f i (s) 2 ds = t 0 n i=1 f i (s) 2 ds = t 0 ∥f (s)∥ 2 ds.Lemma 2 (Tweedie's Formula [12] ). If µ ∼ g(•), z|µ ∼ N (αµ, σ 2 I), and therefore z ∼ f (•), we haveEQUATION)Lemma 3. Suppose y = A(x * ) + ϵ where x * = x 1 (x 0 ) with x 0 being the solution to Eq. (9), A : R n → R m is linear, ϵ ∼ N (0, σ Proof. Recall that the generated auxiliary path y t = α t y + β t Ax 0 . By assumption, we haveA(x t ) = A(α t x + β t x 0 ) = α t A(x(x 0 )) + β t Ax 0 .By subtracting these two equations, we haveEQUATION) As y|x(x 0 ) ∼ N (Ax, σ 2 y I), we have y t |x t ∼ N (Ax t , α 2 t σ 2 y I).The proof for Eq. ( 13) is done. Next, we examine the log probability as follows:EQUATIONEQUATIONTrained by the objective defined in Eq. ( 5), the optimal velocity field would beEQUATION) By defining the signal-to-noise ratio as λ t = α t /β t and rearranging the equation above, we get exactly Eq. ( 11) which we display again below:EQUATION)When α t = t, β t = 1 -t, the equation above becomesEQUATIONA.2 Proof of Theorem 1Before we dive into the proof, we first point out lim ∆t→0 N i=1 γ i = 1. Define the timestep t = (i -1)∆t. Conversely, i = 1 + t/∆t is a function of t. In this sense, we define the i-th step Riemannian discretization of the integral - EQUATIONwhere the decomposition of the second term utilizes the property of the discretization of Riemann integral, and that of the third term utilizes the result in Lemma 3 and thusc i = m 2 log(α 2 i∆t ). By the property of limits, i.e. lim ∆t→0 (N i=1 γ i )( N i=1 ∆p i ) = lim ∆t→0 ( N i=1 γ i ) lim ∆t→0 ( N i=1 ∆p i ) = lim ∆t→0 N i=1 ∆p i ,we can further decompose the second term in Eq. ( 28) into lim ∆t→0 (N i=1 γ i )( N i=1 ∆p i ).By extracting the limit out in Eq. ( 28), the equation becomesEQUATIONEQUATIONwhere Ji := log p(x 0 ) + i j=1 ∆p j + log p(y i∆t |x i∆t ). We further define the c(N ) = N i=1 γ i c ilog p(y).Recall that Ĵi = log p(x (i-1)∆t ) -tr ∂v θ (x (i-1)∆t ,(i-1)∆t) ∂x ∆t + log p(y i∆t |x i∆t ). By triangle inequality, we haveEQUATION⩽ log p(x(x 0 )|y) -EQUATION)Taking the limit on both sides, we haveEQUATIONEQUATIONIn the following, we analyze the two terms on the right-hand side one by one. For the first term: as | • | : R → R is a continuous function, the first term on the right-hand side is equal toEQUATIONEQUATIONEQUATIONwhere the first equation is derived by subtracting the first term in Eq. ( 30) from Eq. ( 26). As the velocity fieldEQUATIONSimilarly, the second term in (38) would beEQUATIONCombining the results in Eq. ( 39) and Eq. ( 40), we getEQUATIONFor the second term: Intuitively, the error between the integral and the Riemannian discretization goes to 0 as ∆t tends to 0. Rigorously,EQUATIONEQUATION)Combining the results of the first term and the second term, we get the proof of theorem 1 done.To quantify our deviation from the assumption of having x t exactly follow the interpolation path α t x + β t x 0 , we define the following: given a differentiable process {z t } and an interpolation path specified by α := {α t } and β := {β t }, we define the trajectory's compliance S α,β ({z t }) to the interpolation path asEQUATIONThis generalizes the definition of straightness in [29] to general interpolation paths. We recover their definition by setting α t = t and β t = 1 -t. In certain cases, we have exact compliance with the predefined interpolation path. For example, when {z t } is generated by v θ and α t = t and β t = 1 -t, note that S α,β ({z t }) = 0 is equivalent to v θ (z t , t) = c where c is a constant, almost everywhere. This ensures that z 1 = z 0 + c. In this case, when generating the trajectory through an ODE solver with starting point x 0 and endpoint x t , we have x t = α t x + β t x 0 , ∀t. When S α,β ({z t }) is not equal to 0, we show in Proposition 2 that we can bound the deviation of our trajectory from the interpolation path using this compliance measure. When specifying our result to Rectified Flow, we can obtain an additional bound showing that when using L-Rectified Flow, the deviation of the learned trajectory from the straight trajectory is bounded by O(1/L).Proposition 2. Consider a differentiable interpolation path specified by α := {α t } and β := {β t }.Then the expected distance between the learned trajectory z t = z 0 + t 0 v θ (z s , s)ds and the predefined trajectory ẑt = z 0 + t 0 ( αs z 1 + βs z 0 )ds can be bounded asEQUATIONIf the differentiable process {z t } is specified by L-Rectified Flow and α t = t and β t = 1 -t for all t ∈ [0, 1], then we additionally haveEQUATIONProof. At time t, we are interested in the distance between a real trajectory z t = z 0 + t 0 v θ (z s , s)ds and a preferred trajectory ẑt = z 0 + t 0 ( αs z 1 -βs z 0 )ds. Using the result in Lemma 1, the distance can be bounded by Therefore,EQUATIONEQUATIONEQUATIONEQUATION) If {z t , t ∈ [0, 1]} is a learned L-rectfied flow, i.e. α t = t and β t = 1 -t in this case, where L is the times of rectifying the flow, by Theorem 3.7 in [29] , we have S α,β ({z}) = O(1/L) and thusEQUATIONEmpirically, [30, 29] found L = 2 generates nearly straight trajectories for high-quality one-step generation. Hence, while this result gives us a simple upper bound, in practice the trajectories may comply more faithfully with the predefined interpolation path than this result suggests.Iteration steps K We provide additional ablation results of K in terms of SSIM in Fig. 6 .NFEs N We first refer to Fig. 1 (c) for a preliminary ablation on N using a toy example. Next, we show PSNR and SSIM scores for varying N in the task of super-resolution. We find that N = 100 is the best trade-off between time and performance. The ablation results are shown in Fig. 7 . In Tab. 3, we present the computational efficiency comparison results. Note that OT-ODE is the slowest as it requires taking the inverse of a matrix r 2 t AA T + σ 2 y I each update time. Our method requires taking the gradient over an estimated trace of the Jacobian matrix, which slows the computation. Experiments were conducted on a Linux-based system with CUDA 12.2 equipped with 4 Nvidia R9000 GPUs, each of them has 48GB of memory.Operators For all the experiments on the CelebA-HQ dataset, we use the operators from [10] . For all the experiments on compressed sensing, we use the operator CompressedSensingOperator defined in the official repository of [14] 4 , Evaluation Metrics are implemented with different Python packages. PSNR is calculated using basic PyTorch operations, and SSIM is computed using the pytorch_msssim package.The workflow begins with using 1,000 FFHQ images at a resolution of 1024×1024. The training process involves 10,000 iterations, each with a batch size of 64, and utilizes the Adam optimizer [26] with a learning rate of 2e-4 and a warmup phase lasting 100 steps. Notably, convergence is achieved within approximately 200 steps. Lastly, the estimated log-likelihood computation for a batch size of 128 takes around 4 minutes and 30 seconds. We show uncured samples generated from the trained models in Fig. 8 .Figure 8 : Generated samples from the flow trained on 10,000 Gaussian samples.In this setting, σ y = 0.001. We use the ncsnpp architecture, training from scratch on 10k images for 100k iterations with a batch size of 50. We set the learning rate to 1 × 10 -2 . Sudden convergence appeared during our training process. We use 2000 warmup steps. Uncured generated images are presented in Fig. 9 .Figure 9 : Generated samples from the flow trained on 10,000 HCP T2w images.OT-ODE As OT-ODE [37] has not released their code and pretrained checkpoints. We reproduce their method with the same architecture as in [29] . We follow their setting and find initialization time t ′ has a great impact on the performance. We use the y init method in their paper. Specifically, the starting point isEQUATIONwhere t ′ is the init time. Note that in the super-resolution task we upscale y with bicubic first. We follow the guidance in the paper and show the ablation results in Fig. 10 and Fig. 11 .We use the following formula to update for each step in the flow:v(x t , y) = v(x t ) + ζ t -∇ xt ∥y -Ax 1 ∥ 2 ,where ζ t is the step size to tune. We refer to DPS for the method to choose ζ t . We set ζ t = η 2∥y-Ax1(xt)∥ . We demonstrate the ablation of η for this baseline in Fig. 12 and Fig. 13 . Note that there is a significant divergence in PSNR and SSIM for the task of inpainting (box). As we observe that artifacts are likely to appear when η ≥ 100, we choose the optimal η = 75 for the best tradeoff.RED-Diff and ΠGDM We use the official repository5 from Nvidia to reproduce the results of RED-Diff and ΠGDM with the pretrained CelebAHQ checkpoint using the architecture of the guided diffusion repository6 from OpenAI.For RED-Diff, the optimization objective is min µ ||y -A(µ)|| 2 + λ(sg(ϵ θ (x t , t) -ϵ)) T µ. Following the implementation of the original paper, we use Adam optimizer with 1,000 steps for all tasks. We choose learning rate lr = 0.25, λ = 0.25 for super-resolution, inpainting(random) and inpainting(box) and lr = 0.5, λ = 0.25 for deblurring as recommended by the paper.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. , with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.This is given by c(N ) := N i=1 γici -log p(y). Please see the proof of Theorem 1 in Appendix A.2.https://github.com/gnobitab/RectifiedFlowhttps://github.com/Sulam-Group/learned-proximal-networks/tree/mainhttps://github.com/NVlabs/RED-diffhttps://github.com/openai/guided-diffusionhttps://deepinv.github.io/deepinv/ 


Decision: The paper has received all positive reviews, with one strongly in favour but with low confidence and the rest borderline accepts. After the discussion phase, reviewers raised their score. Reviewers appreciate the theoretical contribution of integrating flow matching with into MAP for solving inverse problems is a well motivated and interesting approach, the good writeup and improved experimental results. On the other hand, some significant weaknesses also persist, including lack of some important metrics (perceptual quality), limited usefulness of likelihood information, limited number of baselines and experimental datasets and asymptotic error bound usefulness
In light of the positive rebuttal phase that successfully addressed some of the above-mentioned issues and the positive scores and the overall solid contribution, the paper is accepted.

Reviews: review: paper_summary: paper_summary: The paper proposes efficiently recovering MAP estimates by utilizing flow-matching priors. The main proposition in the presented method is to break down the MAP objective into a sum of N local MAP objectives, facilitating a computationally feasible approach that runs in reasonable time. The results are compared to several proposed baselines in terms of distortion (PSNR/SSIM) showing consistent improvements across two datasets and multiple tasks.
strength_weakness: strength_weakness: *    The paper argues that flow-matching priors are useful due to the ability to calculate reconstruction log-likelihood, yet this information is never used and does not appear in the experiments section
*    The authors focus only on distortion (and not on perceptual quality), arguing for a single (blurry) MAP estimate. Nonetheless, by this point it is well understood within the image restoration community that summarizing posteriors to a single prediction is inevitably throwing away uncertainty information which is vital for proper down-the-line decision-making in safety-critical applications. Therefore, posterior samplers (aka stochastic inverse problem solvers) are a much better solution in underdetermined inverse problems with a multitude of admissible solutions. 
*    Nowadays posterior samplers utilizing distilled diffusion models can produce samples with a single neural function evaluation. Compared to such methods, it is unclear what might be the advantage of the presented technique besides maybe accompanying each reconstruction with its likelihood, which is again missing in the presented results.
questions: questions: *    What information is there to be gained from the likelihood computation if all the user is going to be presented with is a single MAP estimate?
*    In terms of runtime, how does your method fair with other types of MAP estimates employing different generative models as priors? 
*    What changes will it take to apply your method to non-linear inverse problems?
limitations: limitations: The authors were upfront about the limitations of their method and stated these in Appendix A.
paper_summary: paper_summary: This paper addresses the challenge of solving linear inverse problems in high-resolution image synthesis using generative models based on flow matching. While these models are appealing due to their ability to compute image likelihoods directly from a learned flow, they suffer from slow log-likelihood computations that involve backpropagation through an ODE solver. This computational bottleneck can be particularly problematic for high-dimensional problems.

To overcome this issue, the authors propose a novel iterative algorithm designed to approximate the maximum-a-posteriori (MAP) estimator efficiently. The key insight is to decompose the MAP objective into multiple "local MAP" objectives, leveraging Tweedie’s formula to perform gradient steps for sequential optimization. This approach allows for a more efficient approximation of the MAP estimator.

The paper validates the proposed algorithm across various linear inverse problems, such as super-resolution, deblurring, inpainting, and compressed sensing. The results show that the new method outperforms existing techniques based on flow matching, making it a promising tool for high-resolution image synthesis and related tasks.

The authors conclude that their approach successfully addresses the computational challenges of flow matching models, offering a robust solution for incorporating flow priors in solving linear inverse problems. They also mention discussing limitations and future work in the appendix.
strength_weakness: strength_weakness: The the paper could benefit from more clear presentation, tying the proofs and experimental results to the claims introduced at the beginning a bit tighter.
questions: questions: NA
limitations: limitations: NA
paper_summary: paper_summary: The paper proposes a flow prior under the MAP structure for solving inverse problems, a theoretical analysis is also given.
strength_weakness: strength_weakness: 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.
questions: questions: N/A
limitations: limitations: N/A
paper_summary: paper_summary: The paper investigates the application of flow matching-based generative models for high-resolution image synthesis, particularly as priors for solving inverse problems. A notable challenge addressed is the slow computation of log-likelihoods in high-dimensional contexts, necessitating backpropagation through an ODE solver. To overcome this, the authors propose an efficient iterative algorithm for approximating the maximum-a-posteriori (MAP) estimator. This method involves approximating the MAP objective through a series of "local MAP" objectives and employs Tweedie's formula for sequential gradient optimization. The proposed method's performance is validated across multiple inverse problems and various datasets.
strength_weakness: strength_weakness: - Insufficient Empirical Evidence: The paper lacks enough qualitative results and empirical evidence to conclusively demonstrate the superiority of the proposed method.
- Need for Additional Metrics: Metrics like FID and LPIPS scores should be included alongside PSNR and SSIM.
- Figure 2 Presentation: The data in Figure 2 should be presented in a table format for easier interpretation of quantitative performance improvements.
- Limited Dataset Testing: For natural image experiments, the algorithm was only tested on the CelebA-HQ dataset. Pretrained models for other datasets (LSUN-bedroom, LSUN-church, and AFHQ-cat) are publicly available from the authors of the rectified flow paper and should be included.
- Blurry reconstruction results:  A closer look at Figure 3 (a,c) reveals that the reconstructed images are too smooth and lack high-frequency details. Compared to OT-ODE, the results are smoother and blurrier.
- Comparison with Recent Work: Apart from OT-ODE, recent work like "D-Flow: Differentiating through Flows for Controlled Generation" also addresses inverse problems and should be considered for comparison.
questions: questions: - Does the proposed method work for very noisy corruptions with high measurement noise (e.g., sigma_y = 0.2 or more)?
- Does the proposed method work for severely ill-posed inverse problems like 8x super-resolution or inpainting with 90% of pixels missing?
- Why was the algorithm evaluated on only 100 images from the CelebA-HQ dataset? Why wasn't the entire validation dataset used?
-There are numerous qualitative results for medical images in the appendix, but for natural image datasets like CelebA-HQ, there is only one small figure (Figure 3). It is hard to draw any conclusions based on just Figure 3. Was the authors' focus primarily on medical images?
limitations: limitations: - The paper's empirical validation is limited, requiring more comprehensive testing across various datasets and additional qualitative results to strengthen the claims of superiority.
- The presentation of quantitative results could be improved by using tables for easier comparison and interpretation.
- The evaluation metrics need to be expanded to include FID and LPIPS scores to provide a more comprehensive assessment of performance.
- Blurry results

Comments: title: A summary of the discussions
Response to the Rebuttal
Kind reminder
comment: Dear AC,

Thank you for your time and effort in evaluating our work. We would like to provide a brief summary of the discussions between the reviewers and us to facilitate a better evaluation. Overall, our responses to each reviewer's questions received positive feedback.

- **Reviewer hhXk:** After acknowledging our method's strength over the recent concurrent work, D-Flow, the reviewer expressed major concerns regarding likelihood information and perceptual quality in their second-to-last comment. We emphasized that our method focuses on computationally efficient, high-quality reconstructions with highly competitive perceptual performance compared to other baselines, as evidenced by our rebuttal PDF. Following our response, the reviewer stated, *raised my score to enhance your chances of getting accepted.*

- **Reviewer sT2G:** The reviewer’s main concerns were about the comparison with traditional methods, more challenging experiment settings, and the need for clearer specifications in our theorem statement. After our response, the reviewer had no further questions on the experiments required and stated, *increase the contribution rating and my confidence.*

- **Reviewer qjx2:** The reviewer acknowledged that we *have addressed my concerns by providing empirical evidence for the requested cases*, such as severely ill-posed inverse problems in highly noisy settings, with more baselines compared and perceptual metrics reported. Consequently, the reviewer chose to *raise the score to 5*.

We hope this summary helps you better evaluate the discussion process.


Best,
Authors
We greatly appreciate the time and effort you've dedicated to our work. Thank you for your support and thought-provoking suggestions!
Thank you for these additional clarifications. After reading the reviews of others, re-reading the paper, and looking again into the rebuttal PDF, I would like to raise my score to 5 to enhance your chances of getting accepted. Best of luck!
We greatly appreciate your time and recognition of our paper's strengths compared to the concurrent work, D-Flow. We would like to provide a last-minute clarification on the two points you mentioned, which we hope will address some of your concerns:

- **Regarding likelihood information:** We appreciate that giving the user likelihood estimates in a more systematic way could be useful, but we want to highlight that the main focus of our work is using a flow-based prior's likelihood to help obtain high-quality reconstruction estimates for inverse problems in a computationally efficient way. As compared to baselines such as D-Flow, we can do this in a computationally efficient manner based on our theoretically-motivated MAP approximation. Moreover, focusing on reconstruction enables a fair comparison with flow-based baselines such as D-Flow and OT-ODE, as well as diffusion-based methods like RED-Diff and $\Pi$GDM. Of note is that likelihood information has also not been provided in the experiments of the four baselines.

- **Perceptual quality:** We would like to highlight that our method demonstrates strong competitiveness in perceptual quality, as evidenced by the fact that our method outperforms $\Pi$GDM in terms of FID in super-resolution and inpainting (random), surpasses OT-ODE in both FID and LPIPS in inpainting (random) and inpainting (box), and exceeds RED-Diff in both FID and LPIPS across all tasks except Gaussian Deblurring. As shown in the Figure 1 in the rebuttal pdf, our method generates clear images even in the highly noisy setting and is strong in terms of PSNR and SSIM. Overall, our method achieves consistently improved performance in reconstruction while maintaining competitive perceptual quality compared to the baselines.
We are glad  to hear that your concerns have been addressed. Thank you for your time and support.
We greatly appreciate your confidence in our work and your thoughtful suggestions regarding our theoretical contributions.   We promise to incorporate the necessary adjustments in the revision.  **Before the discussion period ends,  we would like to provide a last-minute  highlight of  the strengths of our approach, particularly in comparison to the concurrent flow-based method, D-Flow [1], as recognized by other reviewers.**  As shown in the Table 3 in the appendix of our paper, our method requires 1.6 min for each image, OT-ODE requires 2.46 min, while the concurrent work D-Flow [1], which formulates the MAP into a constrained optimization problem (Eq. 9), requires 5-10 mins for each image as documented in their Sec. 3.4. This is because each of its optimization steps requires a backpropogation through an ODE solver, as they calculate the log-likelihood fully. Our work is significantly faster due to our principled local MAP approximation. We hope this runtime comparison further highlights the novelty of our method.  

[1] D-Flow: Differentiating through Flows for Controlled Generation. Heli Ben-Hamu, Omri Puny, Itai Gat, Brian Karrer, Uriel Singer, Yaron Lipman. ICML 2024 (concurrent work)
The authors have addressed my concerns by providing empirical evidence for the requested cases. Hence, I raise my score to 5.
Thanks for the clear and thorough rebuttal. I'm hesitant to change my score mainly because I do not agree with you on two points:
* First, the likelihood information, if the user is to be presented with a single reconstruction, is practically useless as ultimately what matters is the likelihood ratio between different possibilities.
* Second, the results are extra blurry. As evident in the FID numbers from the rebuttal PDF, this method is hardly competitive in perceptual quality. 

I do recognize the advantage of this method over the concurrent work of D-Flow. Nonetheless, in its current form, I think it has limited applicability and therefore I maintain my score.
Dear reviewer,

Thanks a lot for your time and effort. As the discussion period approaches its deadline, we would greatly appreciate your feedback on whether our rebuttal has adequately addressed your concerns. Please reach out if you have additional questions or comments which we are happy to address. Thank you again.

Best, Authors
Dear reviewer,

Thanks a lot for your time and effort. As the discussion period approaches its deadline, we would greatly appreciate your feedback on whether our rebuttal has adequately addressed your concerns. Please reach out if you have additional questions or comments which we are happy to address. Thank you again.

Best,
Authors
Thank the authors for the rebuttal. 

I strongly recommend the authors to characterize more about the non-asymptotic case of c(N), given that the implementation is indeed not exactly the same as the assumption made in asymptotic case. The authors should explicitly clarify this in the paper, especially in the Theorem part. I have no question about the comparison with the traditional methods and the experiment settings. I appreciate the authors for their hard work on the additional experiments and clarification. After careful consideration, I keep the original rating, but increase the contribution rating and my confidence.
Dear all reviewers,

Thank you for all the insightful and constructive comments and feedback, which greatly helped us to improve the paper quality. 

This is a kind reminder that the discussion period is closing in a few days. Feel free to check our response to your questions and tell us if you have any further concerns.

Best,
Authors