Title: Split Gibbs Discrete Diffusion Posterior Sampling
Venue: NeurIPS 2025
Accepted: yes
Keywords: discrete diffusion model

posterior sampling

inverse problem
Abstract: We study the problem of posterior sampling in discrete-state spaces using discrete diffusion models. While posterior sampling methods for continuous diffusion models have achieved remarkable progress, analogous methods for discrete diffusion models remain challenging. In this work, we introduce a principled plug-and-play discrete diffusion posterior sampling algorithm based on split Gibbs sampling, which we call SGDD. Our algorithm enables reward-guided generation and solving inverse problems in discrete-state spaces. We demonstrate the convergence of SGDD to the target posterior distribution and verify this through controlled experiments on synthetic benchmarks. Our method enjoys state-of-the-art posterior sampling performance on a range of benchmarks for discrete data, including DNA sequence design, discrete image inverse problems, and music infilling, achieving more than 30% improved performance compared to existing baselines.


Peer reviews:
comments: SUMMARY:
- Method for sampling from posterior distribution $p(x|y)$ and reward tilted distribution.
    - Introduce an auxiliary variable $z$ and an augmented distribution $\pi$
    - Both marginal of $\pi$ converge to the posterior distribution.
    - Sampling is done by alternating a likelihood sampling step and a prior sampling step.
- They adapt prior work  to discrete diffusion.
- Prove the convergence of their method
- Substantial experiments across challenging tasks

STRENGTHS AND WEAKNESSES:
### Strengths

- The paper is well written and easy to follow. 
Figure 1 illustrates the method well and helps understanding the paper.
- The method does not require gradients w.r.t the likelihood.
- The experiments are substantial, challenging and showcases the performance of the method.
- The theory is convincing and presented clearly.
- It appears the method does not need any additional training.
- The insights regarding noise schedule selection for sampling are valuable and well-explained.

### Weaknesses

- The framework appears designed specifically for uniform transition kernels. Could the authors clarify this limitation? If confirmed, what prevents extension to more general kernels? While absorbing kernels have inherent limitations, they currently represent the state-of-the-art, and demonstrating compatibility would significantly strengthen the work. Additionally, recent literature explores more promising kernels beyond masked processes, and generalization to these newer approaches would enhance the method's broader applicability.

#### Theoretical concerns

I have not read the proof in details but from what I have seen I do not see any theoretical concerns.

QUESTIONS:
See weakness

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-27

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an adaptation of the PnPDM split Gibbs sampler ([1]) to sampling from conditional distributions that use a discrete diffusion generative models as priors. To do so, it proposes using a different coupling potential for the two variables of the split Gibbs that makes the prior step of the Gibbs sampler be the equivalent of the backward part of the generative model, much like the standard gaussian coupling potential does for the continuous split Gibbs sampler in [1]. It then states a theoretical results for fixed $eta$ law of the Gibbs sampler, that is heavily inspired by the theoretical result of [1]. It then proceeds to a numerical verification of the proposed algorithm in several benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper proposes an interesting way of adapting PnPDM to the discrete diffusion case. The approach is sound and the adaptation of the coupling potential is appropriate. The results about the split Gibbs are described in a detailed and precise fashion. \n\nWeaknesses:\n1. The claims in the paper concerning SMC-based methods are highly biased. Namely, lines 190-192. SMC guarantees hold for finite number of particles and I do not see why the guarantees from theorem 1 are inherently better than SMC. Furthermore, the role of the potential functions on SMC can be equated to the choices of $eta$ in the split Gibbs mechanism. Furthermore, most SMC guarantees regard the target posterior distribution and not a $eta$-version as in theorem 1. Therefore, this claim, while not at all necessary to the paper, which could justify their sampler by other merits of the sampler in itself, lead to a biased presentation of the current literature in the topic, which is not acceptable, specially in a prestigious venue as NeurIPS.\n\n2. Numerics pt1: Do the authors try to equate the NFE for all the samplers in table 2 and all section 4? This is on of the only ways of establishing \"fair\" comparison between different samplers, so one would expect so. But, as far as I'm concerned, even in the appendix I cannot find the number of particles used in the SMC approach (except for table 6, which is a different type of experiment).\n\n3. Numerics pt2: Several other small details are missing in the numerics, namely what is exactly the function $G$ in the Monophonic Music problem.\n\nQUESTIONS:\nSee weaknesses for the questions that I had, manly on the numerics part. If some of the writing is improved and details to the numerics are added I am willing to increase my grade.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new diffusion model sampling technique that can be used on discrete target distributions. \nThey augment the space by introducing an auxiliary random variable, $z$, that appears in a \"regularization potential\" term, $D(.)$, controlling the divergence of the intermediate distributions from the target. The augmented posterior is approximated by Gibbs sampling and via updating the original and auxiliary RVs, $x$ and $z$, iteratively. In this context this technique is called \"Split Gibbs\" sampling and is already applied to continuous settings. To apply Split Gibbs sampling to the discrete setting, the current work specifies a new potential function which leads to the new techque, namely Split Gibbs Discrete Diffusion sampling (SGDD). The convergence of the proposed algorithm is proved w.r.t. the Fisher divergecne.\nSGDD is compared with multiple algorithms for discrete diffusion posterior sampling across multiple domains. The reported results show that on these models, SGDD outperforms the existing baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper is well-written; the preliminaries and related work are sufficiently discussed; the SGDD algorithm is explained clearly and compared against several relevant state-of-the-art algorithms on various models.  \n\nWeaknesses: \nMinor comments on the notation:\n1. In lines 68-70 and 140, I assume $\\mathbf{11}^T$?  represents an $N \\times N$ matrix of ones? This notation is a bit confusing and should be clarified.  \n2. Define $\\dot{\\sigma}_t$, (I suppose it denotes $d \\sigma_t /dt$).\n\nPlease also see the Section \"Questions\" below.\n\nQUESTIONS:\n1. In the theorem, the convergence is w.r.t. the Fisher divergence, but in Table 2, the performance is measured w.r.t. Hellinger distance. Could you also report the Fisher divergence as well as the KL divergence from the target? \n\n2. Could you provide a link to the code?\n\nLIMITATIONS:\nThis is a theoretical work with no direct social impact.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed Split Gibbs Discrete Diffusion (SGDD) for posterior sampling in discrete-state spaces, which generalizes the methodology proposed in [1,2,3] that applies continuous diffusion models to posterior sampling problems in continuous-state spaces to discrete distributions. Similar to the procedure developed in [1,2,3], the method consists of two alternative steps. One step is the likelihood sampling step, which is performed via standard Markov Chain Monte Carlo (MCMC) method. The other step is the prior sampling step, which is performed by running a discrete diffusion model. Both theoretical analysis of the proposed algorithm and empirical experiments on a wide range of tasks like DNA design, discrete image inverse problems and music infilling are included to justify the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nPros: This paper studies an important problem (posterior sampling) for the setting of discrete diffusion models, which can be potentially generalized to other problems like inference-time scaling or inference-time alignment. Both theoretical analysis and multiple experiments (along with a wide range of baseline methods) are provided to justify how effective the proposed method is. Moreover, the authors included Python code for the reviewers to test on. \n\nCons: One of the concerns that the reviewer has is that a few work seem to be missing in the literature review. For instance, for diffusion posterior sampling methods based on the sequential monte carlo (SMC) approach, a lot of recent and concurrent work like [4,5,6,7,8,9,10] probably need to be cited and discussed briefly (Though some of them might have focused on different problem settings like inference-time scaling). Similar issues also exist for other classes of work like guidance-based diffusion posterior sampling and theoretical analysis of discrete diffusion models. Hence, the authors are encouraged to perform a more thorough review of related methods developed in these subfields. \n\nFurthermore, for the theoretical analysis part, the reviewer thinks the assumptions given in Appendix A might be a bit too strong, as they essentially assume that the score function can be ideally learnt with respect to the $l_{\\infty}$ norm. Compared to other existing work on the theoretical analysis of discrete diffusion models like [12,13,14,15,16], the assumptions might be too strong. The reviewer will elaborate on such issue in the \"Questions\" section below. \n\nMoreover, for imaging inverse problems considered in this manuscript, the authors only tested on the MNIST dataset. In contrast, related work like [1,2,17] have tested on large-scale image datasets like FFHQ [18] and ImageNet [19]. The authors are encouraged to test the proposed methodology on these large-scale datasets to justify the effectiveness of the proposed method.  \n\nAdditionally, several issues and typos also exist regarding the presentation of this paper. For instance, the term $dt$ is missing for almost all integrals of the relative Fisher information appearing in the manuscript (Examples include line 181-182, line 567-570). Also, definition of $\\mu_t$ and $\\pi_t$ included in line 174-177 seems to be a bit ambiguous. The authors may consider including a similar picture like Figure 3 in [1] to better illustrate the relation between these two processes (or maybe simply referring to Figure 3 in [1]).\n\nQUESTIONS:\nRegarding the theoretical part presented in the manuscript, would it be possible for the authors to comment on the relation between the relative Fisher information FI defined here with the score entropy loss proposed in [21]? It seems to the reviewer that the two forms are akin and highly related. Instead of adopting the stronger assumption on the error between the true score function and the learnt score function with respect to the $l_{\\infty}$ norm, the reviewer thinks that it is probably better to assume an upper bound on the training loss directly, just as what has been done in [13]. Moreover, it seems that the most crucial part in the theoretical results derived here is essentially bounding the KL divergence between two continuous-time markov chains (CTMCs) $\\frac{\\partial}{\\partial t}\\mu_t = Q_t \\mu_t$ and $\\frac{\\partial}{\\partial t}\\widehat{\\mu}_t = \\widehat{Q}_t \\widehat{\\mu}_t$ via the discrepancy between $Q_t$ and $\\widehat{Q}_t$, which highly relates to the Girsanov Theorem for CTMCs (See for instance Theorem III.5.34 in the book [11] or Theorem 3.3 in [13] for the case of discrete diffusion models). Hence, the authors might consider citing these related work and briefly comment on the relation between all different derivations. \n\nIn addition, to the best of the reviewer's knowledge, recent work on solving inverse problems via continuous diffusion models have also adopted advanced solvers like [19] to speed up the inference process. Would it be possible for the authors to comment on whether it would be possible for SGDD to be accelerated via more efficient inference methods (for discrete diffusion models) like distillation [22] or high-order solvers [16]?\n\nLIMITATIONS:\nThe reviewer is satisfied with the limitations that the authors briefly discussed at the end of the manuscript. For instance, the authors mentioned that currently the methodology can only be used for discrete diffusion models with uniform transition matrix. Hence, one of the future directions is to investigate whether the methodology can be generalized to other forms of discrete diffusion models like the masked discrete diffusion model with absorbing transition matrix. However, this is indeed an important issue as masked discrete diffusion models (akin to autoregressive models) are more widely adopted nowadays in other tasks. \n\nOverall, the reviewer acknowledges that the authors address an important problem and offer meaningful contributions. However, the manuscript’s quality (presentation, literature review, experiments and theoretical analysis) probably requires further improvement before being considered for publication at a top-tier machine learning venue.\n\nReferences:\n\n[1] Wu, Z., Sun, Y., Chen, Y., Zhang, B., Yue, Y., & Bouman, K. (2024). Principled probabilistic imaging using diffusion models as plug-and-play priors. Advances in Neural Information Processing Systems (NeurIPS), 37, 118389-118427.\n\n[2] Xu, X., & Chi, Y. (2024). Provably robust score-based diffusion posterior sampling for plug-and-play image reconstruction. Advances in Neural Information Processing Systems (NeurIPS), 37.\n\n[3] Coeurdoux, F., Dobigeon, N., & Chainais, P. (2024). Plug-and-play split Gibbs sampler: embedding deep generative priors in Bayesian inference. IEEE Transactions on Image Processing.\n\n[4] Singhal, R., Horvitz, Z., Teehan, R., Ren, M., Yu, Z., McKeown, K., & Ranganath, R. (2025). 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Di $\\mathtt {[M]} $ O: Distilling Masked Diffusion Models into One-step Generator. arXiv preprint arXiv:2503.15457.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-08"}