Discrete Langevin-Inspired Posterior Sampling

2026-05-09

AI Summary

Inverse problems in discrete state spaces have long been constrained by reliance on continuous relaxations, Gibbs updates, or specific noise assumptions, limiting both generality and scalability. This work proposes ΔLPS, the first gradient-guided parallel posterior sampler operating entirely in discrete space, eliminating the need for continuous relaxation and compatible with diverse discrete prior training paradigms such as masked and uniform diffusion. By integrating discrete gradient estimation with a parallel token updating strategy, ΔLPS effectively handles linear, nonlinear, and blind inverse problems. Experiments demonstrate that ΔLPS significantly outperforms existing discrete sampling methods on image restoration and spatial mapping tasks across MNIST, CIFAR, and FFHQ benchmarks, achieving performance on par with state-of-the-art continuous diffusion solvers.

Abstract

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.

Problem

• discrete inverse problems
• posterior sampling
• discrete diffusion models
• discrete state spaces
• gradient-based sampling

Innovation

• discrete diffusion
• posterior sampling
• inverse problems
• gradient-based discrete updates
• Langevin-inspired sampling