Constrained Particle Seeking: Solving Diffusion Inverse Problems with Just Forward Passes

Pretrained diffusion models are effective priors for Bayesian inverse problems, but posterior sampling with these priors is often costly because data-consistency guidance is applied throughout the full reverse trajectory. Existing methods have shown that vector-Jacobian products through the denoiser can sometimes be avoided, yet they typically still rely on dense guidance through the full trajectory or expensive inner solves. We introduce Sparse Scheduled Diffusion Guidance for Inverse Problems (Spin), a solver that avoids starting posterior sampling from pure noise. Spin first samples from a posterior time-marginal at an intermediate timestep $t_*$, and then uses that state as a warm start for a guided reverse diffusion process. At guidance time, instead of enforcing the measurement constraint at every denoising step, Spin applies lightweight corrections only at scheduled timesteps where the denoiser can still clean up artifacts. The resulting procedure decouples prior refinement from data consistency: the prior supplies denoising, while lightweight pixel-space optimization enforces the measurement constraint without backpropagation through the denoiser or decoder. Across linear and nonlinear inverse problems on FFHQ and ImageNet, Spin achieves competitive reconstruction quality with a substantially better runtime--memory profile, running 2x faster on pixel-space models and up to 50x faster on latent diffusion models, with lower memory costs.

We study posterior sampling for inverse problems in discrete state spaces using discrete diffusion models as generative priors. While continuous diffusion models have become widely used for inverse problems, their discrete counterparts remain comparatively underexplored. Existing discrete posterior samplers often rely on continuous relaxations of discrete variables, Gibbs-style updates, or mechanisms specialized to particular corruption processes, which can limit scalability or generality. We propose $Δ$LPS, a Discrete Langevin-Inspired Posterior Sampler that uses gradient information to identify promising discrete moves without leaving the discrete state space. The resulting approach enables efficient parallel updates across all token dimensions and is agnostic to the training paradigm of the discrete diffusion prior, including masked and uniform-state diffusion. We evaluate our method on image restoration tasks across MNIST, CIFAR, and FFHQ, as well as spatial mapping, covering linear, nonlinear, and blind inverse problems. Across these settings, we improve over recent discrete diffusion posterior samplers and are competitive with strong continuous diffusion-based inverse solvers. Our results suggest that fully discrete, gradient-informed posterior samplers offer a scalable and general path toward solving inverse problems over discrete representations.

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing work that combines DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior and the empirical distribution of the generated samples can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods. Our code is available at the following Github repository~https://github.com/HaoxuanSteveC00/AFDPS-TMLR.