Yazid Janati

Badr Moufad, Navid Bagheri Shouraki, Alain Oliviero Durmus et al.

Diffusion models have emerged as powerful priors for image editing tasks such as inpainting and local modification, where the objective is to generate realistic content that remains consistent with observed regions. In particular, zero-shot approaches that leverage a pretrained diffusion model, without any retraining, have been shown to achieve highly effective reconstructions. However, state-of-the-art zero-shot methods typically rely on a sequence of surrogate likelihood functions, whose scores are used as proxies for the ideal score. This procedure however requires vector-Jacobian products through the denoiser at every reverse step, introducing significant memory and runtime overhead. To address this issue, we propose a new likelihood surrogate that yields simple and efficient to sample Gaussian posterior transitions, sidestepping the backpropagation through the denoiser network. Our extensive experiments show that our method achieves strong observation consistency compared with fine-tuned baselines and produces coherent, high-quality reconstructions, all while significantly reducing inference cost. Code is available at https://github.com/YazidJanati/ding.

Samson Gourevitch, Alain Durmus, Eric Moulines et al.

Learning models with categorical variables requires optimizing expectations over discrete distributions, a setting in which stochastic gradient-based optimization is challenging due to the non-differentiability of categorical sampling. A common workaround is to replace the discrete distribution with a continuous relaxation, yielding a smooth surrogate that admits reparameterized gradient estimates via the reparameterization trick. Building on this idea, we introduce ReDGE, a novel and efficient diffusion-based soft reparameterization method for categorical distributions. Our approach defines a flexible class of gradient estimators that includes the Straight-Through estimator as a special case. Experiments spanning latent variable models and inference-time reward guidance in discrete diffusion models demonstrate that ReDGE consistently matches or outperforms existing gradient-based methods. The code will be made available at https://github.com/samsongourevitch/redge.

Samson Gourevitch, Yazid Janati, Dario Shariatian et al.

Discrete diffusion models are often trained through clean-data prediction, but the prediction can be used in different ways to define the reverse dynamics. In Masked Diffusion Models (MDM) these choices largely coincide, whereas in Uniform Diffusion Models (UDM) they do not. We show that the standard plug-in bridge parameterization for UDM is not optimized by the denoising posterior, but by a leave-one-out posterior that predicts each clean token without using its own noisy observation. This identifies a mismatch between the plug-in ELBO and the usual cross-entropy denoising objective. We characterize the leave-one-out target and derive exact conversions between the denoiser, the leave-one-out posterior, and the score. These conversions allow us to disentangle parameterization and training objective. Our results also lead to inference improvements without any additional training through an informed predictor-corrector sampler and improved temperature sampling based on the leave-one-out predictor. We further introduce an absorbing-state reformulation of uniform diffusion that preserves the UDM joint law while decomposing it into masked-diffusion-like sampling operations, with simpler denoising posteriors, carry-over unmasking, and a natural remasking mechanism. On language modeling, leave-one-out parameterizations consistently improve UDM generation, while the absorbing construction matches or surpasses masked diffusion. These results suggest that the empirical gap between masked and uniform diffusion is driven less by the choice of marginals themselves than by parameterization and sampling design. The code and models can be found at https://github.com/samsongourevitch/rev_udm.

Yazid Janati, Badr Moufad, Alain Durmus et al.

Recent advancements in solving Bayesian inverse problems have spotlighted denoising diffusion models (DDMs) as effective priors. Although these have great potential, DDM priors yield complex posterior distributions that are challenging to sample. Existing approaches to posterior sampling in this context address this problem either by retraining model-specific components, leading to stiff and cumbersome methods, or by introducing approximations with uncontrolled errors that affect the accuracy of the produced samples. We present an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior. Our method significantly reduces the approximation error associated with current techniques without the need for retraining. We demonstrate the versatility and effectiveness of our approach for a wide range of Bayesian inverse problems. The code is available at \url{https://github.com/Badr-MOUFAD/dcps}

Badr Moufad, Yazid Janati, Lisa Bedin et al.

Diffusion models have recently shown considerable potential in solving Bayesian inverse problems when used as priors. However, sampling from the resulting denoising posterior distributions remains a challenge as it involves intractable terms. To tackle this issue, state-of-the-art approaches formulate the problem as that of sampling from a surrogate diffusion model targeting the posterior and decompose its scores into two terms: the prior score and an intractable guidance term. While the former is replaced by the pre-trained score of the considered diffusion model, the guidance term has to be estimated. In this paper, we propose a novel approach that utilises a decomposition of the transitions which, in contrast to previous methods, allows a trade-off between the complexity of the intractable guidance term and that of the prior transitions. We validate the proposed approach through extensive experiments on linear and nonlinear inverse problems, including challenging cases with latent diffusion models as priors. We then demonstrate its applicability to various modalities and its promising impact on public health by tackling cardiovascular disease diagnosis through the reconstruction of incomplete electrocardiograms. The code is publicly available at \url{https://github.com/yazidjanati/mgps}.

Badr Moufad, Albina Ilina, Hai Victor Habi et al.

Commercial Microwave Links (CMLs) offer dense spatial coverage for rainfall sensing but produce path-integrated measurements that make accurate ground-level reconstruction challenging. Existing methods typically oversimplify CMLs as point sensors and neglect line integration relating rainfall to signal attenuation, resulting in degraded performance under heterogeneous precipitation. In this work, we view rain field reconstruction as a Bayesian inverse problem with Diffusion Models (DMs) as high-fidelity spatial priors. We show that diffusion models better preserve key rainfall statistics compared to censored Gaussian processes. Framing rainfall estimation as a Bayesian inverse problem with a DM prior enables training-free posterior sampling using a broad family of methods, including Plug-and-Play, Sequential Monte Carlo, and Replica Exchange methods. Experiments on synthetic and real-world datasets demonstrate consistent improvements over established CML-based reconstruction baselines.

Yazid Janati, Badr Moufad, Mehdi Abou El Qassime et al.

Denoising diffusion models have driven significant progress in the field of Bayesian inverse problems. Recent approaches use pre-trained diffusion models as priors to solve a wide range of such problems, only leveraging inference-time compute and thereby eliminating the need to retrain task-specific models on the same dataset. To approximate the posterior of a Bayesian inverse problem, a diffusion model samples from a sequence of intermediate posterior distributions, each with an intractable likelihood function. This work proposes a novel mixture approximation of these intermediate distributions. Since direct gradient-based sampling of these mixtures is infeasible due to intractable terms, we propose a practical method based on Gibbs sampling. We validate our approach through extensive experiments on image inverse problems, utilizing both pixel- and latent-space diffusion priors, as well as on source separation with an audio diffusion model. The code is available at https://www.github.com/badr-moufad/mgdm

Jean Pachebat, Giovanni Conforti, Alain Durmus et al.

We introduce iterative tilting, a gradient-free method for fine-tuning diffusion models toward reward-tilted distributions. The method decomposes a large reward tilt $\exp(λr)$ into $N$ sequential smaller tilts, each admitting a tractable score update via first-order Taylor expansion. This requires only forward evaluations of the reward function and avoids backpropagating through sampling chains. We validate on a two-dimensional Gaussian mixture with linear reward, where the exact tilted distribution is available in closed form.

Léon Zheng, Thomas Hirtz, Yazid Janati et al.

Zero-shot diffusion posterior sampling offers a flexible framework for inverse problems by accommodating arbitrary degradation operators at test time, but incurs high computational cost due to repeated likelihood-guided updates. In contrast, previous amortized diffusion approaches enable fast inference by replacing likelihood-based sampling with implicit inference models, but at the expense of robustness to unseen degradations. We introduce an amortization strategy for diffusion posterior sampling that preserves explicit likelihood guidance by amortizing the inner optimization problems arising in variational diffusion posterior sampling. This accelerates inference for in-distribution degradations while maintaining robustness to previously unseen operators, thereby improving the trade-off between efficiency and flexibility in diffusion-based inverse problems.

Badr Moufad, Yazid Janati, Alain Durmus et al.

Classifier-Free Guidance (CFG) is a widely used technique for improving conditional diffusion models by linearly combining the outputs of conditional and unconditional denoisers. While CFG enhances visual quality and improves alignment with prompts, it often reduces sample diversity, leading to a challenging trade-off between quality and diversity. To address this issue, we make two key contributions. First, CFG generally does not correspond to a well-defined denoising diffusion model (DDM). In particular, contrary to common intuition, CFG does not yield samples from the target distribution associated with the limiting CFG score as the noise level approaches zero -- where the data distribution is tilted by a power $w \gt 1$ of the conditional distribution. We identify the missing component: a Rényi divergence term that acts as a repulsive force and is required to correct CFG and render it consistent with a proper DDM. Our analysis shows that this correction term vanishes in the low-noise limit. Second, motivated by this insight, we propose a Gibbs-like sampling procedure to draw samples from the desired tilted distribution. This method starts with an initial sample from the conditional diffusion model without CFG and iteratively refines it, preserving diversity while progressively enhancing sample quality. We evaluate our approach on both image and text-to-audio generation tasks, demonstrating substantial improvements over CFG across all considered metrics. The code is available at https://github.com/yazidjanati/cfgig

Abduragim Shtanchaev, Albina Ilina, Yazid Janati et al.

Pretrained diffusion models serve as effective priors for Bayesian inverse problems. These priors enable zero-shot generation by sampling from the conditional distribution, which avoids the need for task-specific retraining. However, a major limitation of existing methods is their reliance on surrogate likelihoods that require vector-Jacobian products at each denoising step, creating a substantial computational burden. To address this, we introduce a lightweight likelihood surrogate that eliminates the need to calculate gradients through the denoiser network. This enables us to handle diverse inverse problems without backpropagation overhead. Experiments confirm that using our method, the inference cost drops dramatically. At the same time, our approach delivers the highest results in multiple tasks. Broadly speaking, we propose the fastest and Pareto optimal method for Bayesian inverse problems.

Ahmed Ghorbel, Badr Moufad, Navid Bagheri Shouraki et al.

Text-driven image and video editing can be naturally cast as inpainting problems, where masked regions are reconstructed to remain consistent with both the observed content and the editing prompt. Recent advances in test-time guidance for diffusion and flow models provide a principled framework for this task; however, existing methods rely on costly vector--Jacobian product (VJP) computations to approximate the intractable guidance term, limiting their practical applicability. Building upon the recent work of Moufad et al. (2025), we provide theoretical insights into their VJP-free approximation and substantially extend their empirical evaluation to large-scale image and video editing benchmarks. Our results demonstrate that test-time guidance alone can achieve performance comparable to, and in some cases surpass, training-based methods.

Yazid Janati, Alain Durmus, Jimmy Olsson et al.

Diffusion models enable the synthesis of highly accurate samples from complex distributions and have become foundational in generative modeling. Recently, they have demonstrated significant potential for solving Bayesian inverse problems by serving as priors. This review offers a comprehensive overview of current methods that leverage \emph{pre-trained} diffusion models alongside Monte Carlo methods to address Bayesian inverse problems without requiring additional training. We show that these methods primarily employ a \emph{twisting} mechanism for the intermediate distributions within the diffusion process, guiding the simulations toward the posterior distribution. We describe how various Monte Carlo methods are then used to aid in sampling from these twisted distributions.

Achille Thin, Yazid Janati, Sylvain Le Corff et al.

Sampling from a complex distribution $π$ and approximating its intractable normalizing constant Z are challenging problems. In this paper, a novel family of importance samplers (IS) and Markov chain Monte Carlo (MCMC) samplers is derived. Given an invertible map T, these schemes combine (with weights) elements from the forward and backward Orbits through points sampled from a proposal distribution $ρ$. The map T does not leave the target $π$ invariant, hence the name NEO, standing for Non-Equilibrium Orbits. NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS estimators of expectations under $π$ while NEO-MCMC combines multiple NEO-IS estimates of the normalizing constant and an iterated sampling-importance resampling mechanism to sample from $π$. For T chosen as a discrete-time integrator of a conformal Hamiltonian system, NEO-IS achieves state-of-the art performance on difficult benchmarks and NEO-MCMC is able to explore highly multimodal targets. Additionally, we provide detailed theoretical results for both methods. In particular, we show that NEO-MCMC is uniformly geometrically ergodic and establish explicit mixing time estimates under mild conditions.