A Gibbs posterior sampler for inverse problem based on prior diffusion model

TL;DR

The paper tackles Bayesian inversion with a linear observation model and an ill-posed prior modeled by a diffusion process learned from a large dataset. It proposes a Gibbs sampling strategy (G-DPS) that exploits Gaussian conditional posteriors for both the latent diffusion states and the image of interest, enabling efficient sampling via FFT-based and linear-Gaussian updates. Empirical results on MNIST-based deconvolution show accurate posterior means and reliable uncertainty quantification, with significant computational efficiency and scalability advantages. The approach offers a practical, convergent-compatible framework for diffusion-prior inverse problems, while pointing to future work in parameter estimation and comparative benchmarks.

Abstract

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a prior density, and (3) the latter is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one. This paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that this approach is particularly effective and remarkably simple. In addition, it offers a guarantee of convergence in a clearly identified situation. The results are clearly confirmed by numerical simulations.

Figures (6)

• Figure 1: Hierarchy: ${\mathb{x}}\xspace_{ 0}$ is the image of interest, ${\mathb{x}}\xspace_{ 1:T}$ are the latent images and ${\mathb{y}}\xspace$ is the measured image (blurred and noisy version of the true ${\mathb{x}}\xspace_{ 0}$).
• Figure 2: Samples provided by the Gibbs algorithm for three pixels of ${\mathb{x}}\xspace_{ 0}$. They are shown as a function of iteration index (left) and as histograms (right). They are samples of one dimensional marginal pdfs. See also Fig. \ref{['Fig:ResultsClouds']} and Tab. \ref{['Tab:ParamEstimate']}. The green lines / dots give the true value.
• Figure 3: Point clouds for two dimensional marginals pdfs for three pixels. The sample are given in blue and the true value is given in green. See also Fig. \ref{['Fig:ResultsChains']} for one dimensional plots and Tab. \ref{['Tab:ParamEstimate']} for quantitative assessment.
• Figure 4: Left to right: true object ${\mathb{x}}\xspace^\star$, measurements ${\mathb{y}}\xspace$ and estimated object $\widehat{{\mathb{x}}\xspace}$. The figure shows the images themselves (top) and cross-sections (bottom).
• Figure 5: Cross-sections of ${\mathb{x}}\xspace^\star$ (plain green) and the "uncertainty" intervals (dashed blue) with center the estimate and of width two standard deviations.