Title: Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion Models

Inverse Problems

Posterior Sampling
Abstract: Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood $p(\boldsymbol{y} \mid \boldsymbol{x})$, often resulting in artifacts and instability under complex or high-noise conditions.
To address these limitations, we propose a novel framework called coupled data and measurement space diffusion posterior sampling (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space $\{\boldsymbol{y}_t\}$, evolving in parallel with the data-space diffusion $\{\boldsymbol{x}_t\}$, which enables the derivation of a closed-form posterior $p(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{y}_{t-1})$. This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\n## Summary\n\n**C-DPS** is proposed for  solving linear inverse problems using diffusion models. Unlike prior works that either project generated samples onto the measurement-consistent set or approximate the likelihood gradient, C-DPS introduces a coupled forward diffusion process in both data and measurement space. This enables the derivation of a closed-form posterior at each reverse diffusion step, leading to consistent  Bayesian sampling. The authors provide empirical results  demonstrating  strong performance on several inverse problems, including inpainting, deblurring, and super-resolution.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. Paper is well-motivated, well-presented and clearly written. \n\n2. The idea of adding a parallel diffusion process in the measurement space is both novel and elegant. It allows for better measurement consistency and principled posterior sampling.\n\n3. The experiments are thorough. Both the visual and quantitative results show consistent improvements over strong baseline methods like DPS, ReSample, and ΠGDM.\n\n## Weaknesses \n\n1. The algorithm 2 is not entirely faithful to the proposed idea. While the paper introduces a diffusion process for the measurements, the actual algorithm seems to bypass this and relies mostly on Step 16 for computation.. \n\n2. Although the claim is that  the method is efficient (line 79), no actual runtime comparisons or measurements is provided to back this up.\n\n3. It’s unclear why PSNR, a standard metric for evaluating inverse problem solutions, is not included in the reported results. This would help compare more directly with other existing methods.\n\nQUESTIONS:\n1. The authors could improve the algorithm to better reflect their proposed method. \n2. The authors could add PSNR and also time to better reflect their contribution (This would definitely motivate this reviewer to increase the score).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an interesting approach for diffusion posterior sampling to sample from posterior distributions with diffusion-trained priors. The idea is to build noisy sequence of observations, instead of noising in x-space as usually done, instead deriving the kernel $x_{t-1} | x_t$ based on the noising process of data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- This is an interesting and novel idea as it is usually hard to derive noising processes for the data - and in the literature this is usually done heuristically. This is a more principled approach which seem to perform well.\n- The paper is written well and efficiently, it provides the derivations cleanly.\n- This approach can be potentially important for further extensions, e.g., various approximations of $p(x_{t-1} | x_t, y_t)$.\n\nWeaknesses: Insufficient comparison, especially to more than one relevant method in the literature. See my questions for details, I think it is crucial that these weaknesses are addressed.\n\nQUESTIONS:\nI am generally happy with this paper - I think it gives a solid approach and definitely an extensive experimental exploration. I have a few questions.\n\n1) Constructing artificial data sequences for diffusion posterior sampling is proposed in the literature before: citations [25] and [35] in your paper, among others. However, these are not added in the comparison. There is no 'related work' section for the precise differences between your data noising approach and these works. I know that there is a related work section in the appendix but this does not single out approaches with artificial data sequences. Please write a summary paragraph with all works which create a sequence $(y_t)_{t=0}^T$ specifically, in the main text.\n\n2) Why are approaches from [16], [25], and [35] not compared to? Especially since they are used in experimental setting. You should definitely include [16] as a comparison, as your experimental setting is very similar. For example, you would obtain comparable performance in Figure 3 - and in some image experiments -- in this way, the comparison seems to be made only to the weaker methods.\n\n3) Can you clarify the sentence in line 183? I understand freezing the network is because you want a Gaussian distribution. But I don't know if there is any other issue with evaluating the score otherwise. I wonder, in a simple example, how an empirical (sampling-based) approach of (13) would perform in comparison to the Gaussian approximation using the frozen score.\n\nSmall notes:\n\n- the reference [16] (TMLR) and [28] (NeurIPS) are published - please correct your citations for similar arXiv vs published paper mistakes.\n\nLIMITATIONS:\nYes; but in the appendix.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper aims to eliminate projection and likelihood-approximation steps during diffusion sampling for inverse problem solving. It introduces an auxiliary Markov chain for the measurement $y$, defining the marginal distribution at time $t$ as $p(y_t|x_t)$l this contrasts with prior works that relies on $p(y_0|x_t)$.\n\nTo sample $y_t$ from $p(y_t|x_t)$, the paper derives a closed-form update that is coupled with the diffusion update for $x_t$. The implementation assumes that the score function evaluated at $x_{t-1}$ can be replaced by the score function at $x_t$.\n\nExperiments show that the proposed method improves inverse-problem performance relative to the chosen baselines.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper proposes a novel idea on introducing auxiliary Markov chain for the measurement.\nHowever, the paper contains logical flaws which limits its quality and significance. Also, experiments have done with old baselines and the paper omits recent works on inverse problem solving beyond projection-based or likelihood approximation-based solvers, which also limits the quality and significance. Please see the below list for details.\n\n**Strength**\n- The paper provides all methodological details.\n- Although most baselines are outdated, the experiments cover a wide range of inverse problems across two datasets.\n\n**Weakness**\n- Technical error in Eq (9): the term $p(x_{t-1})$ is omitted without any reason. In addition, the factorization $p(x_t,y_{t-1}|x_{t-1}) = p(x_t|x_{t-1})p(y_{t-1}|x_{t-1})$ assumes independence between $y_{t-1}$ and $x_t$ (not described in the paper), which conflicts with Eq (15), where the score at $x_{t-1}$ is simple replaced by the score at $x_t$.\n- No sufficient discussion or theoretical validation on the effect of coupling stochastic update of $y_t$ with deterministic update of $x_t$.\n- Assumption in Eq (14) limits efficiency, as the resulting error becomes significant when fast samples are used - even with DDIM.\n- The appendix contains no qualitative examples for the medical-image domain, and also provided qualitative results in the main body is limited, making visual assessment difficult.\n- Important non-projection and non-likelihood methods [1-4] are missing from the introduction and related works.\n- The evaluation omits more recent approaches such as [1-6]\n\n\n**Referece**\n\n [1] A Variational Perspective on Solving Inverse Problems with Diffusion Models, arxiv 2023\n \n [2] Repulsive Latent Score Distillation for Solving Inverse Problems, ICLR 2025\n \n [3] Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing, CVPR 2025\n \n [4] Inverse Problem Sampling in Latent Space Using Sequential Monte Carlo, ICML 2025\n \n [5] Prompt-tuning latent diffusion models for inverse problems, ICML 2024\n \n [6] Denoising Diffusion Models for Plug-and-Play Image Restoration, CVPRW2023\n\nQUESTIONS:\n- Please provide theoretical clarification on, especially, equations (9), (14), and (15). If I mis-understood something, please point out it.\n- Please provide comparison result with more recent works, provided in the weakness section.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the intractable likelihood problem in diffusion posterior sampling by introducing a coupled noisy measurement trajectory. Specifically, during the sampling process, the method constructs a coupled measurement sequence $\\\\{\\boldsymbol{y}\\_t\\\\}\\_{t=0}^T$ following the standard diffusion forward process. Subsequently, it samples $\\boldsymbol{x}_t$ from the conditional posterior $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ iteratively in reverse time order. Utilizing Tweedie's formula along with mild assumptions about the score function, the conditional posterior $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ can be analytically derived as a Gaussian distribution. The proposed approach demonstrates superior performance on various image restoration tasks and a synthetic benchmark dataset designed to provide ground-truth posterior distributions.\n\nSTRENGTHS AND WEAKNESSES:\n> Strengths\n\nThe paper is well motivated, clearly written, and easy to follow. The key strengths of this paper are summarized as follows:\n\n1. **Combination of Tweedie's Approximation with Noisy Measurement.** The use of Tweedie's approximation $\\boldsymbol{x}_0(\\boldsymbol{x}_t)$ can lead to blurred results for larger time steps $t$; however, the method compensates for this issue through the noisy measurement $\\boldsymbol{y}\\_{t}$ as expressed in Equation (10).\n\n2. **Superior Empirical Performance.** The proposed method consistently outperforms existing techniques on standard image restoration benchmarks and provides detailed analyses supported by synthetic experiments.\n\n3. **Clear Visualization of Measurement Error Evaluation.** Figure 4 effectively highlights the method's capability to closely track measurement signals throughout the sampling process, demonstrating clear empirical superiority.\n\n> Weaknesses\n\nHowever, the paper also has a few weaknesses:\n\n1. **Similar Ideas in the Literature.** The idea of establishing a diffusion process in the measurement space is not new. It was first introduced in Appendix I.4 of [1] and then investigated in [2] for medical imaging problems. The authors should cite [2] and compare the proposed method with the method in [2].\n1. **Restriction to Linear Inverse Problems.** The approximated closed-form solution for the distribution $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ relies on a linear forward model. It would be valuable to explore or discuss potential extensions to nonlinear inverse problems. Does the author have any insight on this?\n2. **Limited Baseline Comparisons on Measurement Fidelity.** While the paper demonstrates lower measurement errors and better reconstruction results compared to selected baselines, it lacks comparisons with more recent methods with improved measurement fidelity, such as DDNM [3], PnP-DM [4], DCDP [5], DAPS [6], SITCOM [7], and DMPlug [8]. The author should compare with some of them for better accessing the performance.\n3. **Absence of Common Evaluation Metrics.** The paper does not report PSNR values, which is a standard metric widely used and reported by many baseline studies. The authors should include it as well if possible?\n\n[1] Song et al., Score-Based Generative Modeling through Stochastic Differential Equations\n\n[2] Song et al., Solving Inverse Problems in Medical Imaging with Score-Based Generative Models\n\n[3] Wang et al., Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model\n\n[4] Wu et al., Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors\n\n[5] Li et al., Decoupled Data Consistency with Diffusion Purification for Image Restoration\n\n[6] Zhang et al., Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing\n\n[7] Akhouri et al., SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems\n\n[8] Wang et al., DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models\n\nQUESTIONS:\nPlease refer to the weaknesses part for my main questions. Here's several additional questions.\n\n1. **Samping time**. The pre-whitened conjugate gradient is only practical when forward function is efficient. If computing the forward function is costly, the sampling step becomes time-consuming. Could the authors provide some insights or strategies for addressing this limitation?\n\n2. **Sampling diversity.** In 92% random inpatining tasks, the measurement signal is extremely sparse, which means the reconstructions can exhibit diverse appearance yet still fit the measurement equally well. Does the author observe this property, or do the reconstructions always consistently resemble the ground truth, as illustrated in Figure 2?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}