Discrete Langevin-Inspired Posterior Sampling

TL;DR

This paper introduces $ riangle$LPS, a novel discrete Langevin-inspired posterior sampler that efficiently utilizes gradient information for inverse problems in discrete spaces, outperforming existing methods.

Contribution

The paper presents a scalable, gradient-based discrete posterior sampling method compatible with various diffusion priors and inverse problem types.

Findings

$ riangle$LPS outperforms recent discrete diffusion posterior samplers.

The method is competitive with continuous diffusion-based inverse solvers.

It is effective across multiple inverse problem settings and datasets.

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 $\Delta$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.