Solving Linear-Gaussian Bayesian Inverse Problems with Decoupled Diffusion Sequential Monte Carlo

TL;DR

This work tackles Bayesian inference for linear-Gaussian inverse problems where the prior is defined implicitly by a pre-trained diffusion model and the likelihood is Gaussian in y with y = Ax + noise. It introduces Decoupled Diffusion SMC (DDSMC), combining a generalized DAPS prior with diffusion-based likelihoods to permit large, decoupled sample updates while preserving asymptotic exactness; the reconstruction f_theta and PF-ODE perspectives are leveraged to design effective proposals. The authors provide explicit formulations for diagonal and non-diagonal measurement operators A, extend the framework to discrete data via D3SMC, and demonstrate strong performance on synthetic Gaussian mixtures, image restoration tasks, and protein structure completion, with qualitative results on discrete MNIST. Overall, the method offers a flexible, asymptotically exact posterior sampler that bridges high-dimensional inverse problems and diffusion priors, with practical applicability to continuous and discrete data domains.

Abstract

A recent line of research has exploited pre-trained generative diffusion models as priors for solving Bayesian inverse problems. We contribute to this research direction by designing a sequential Monte Carlo method for linear-Gaussian inverse problems which builds on "decoupled diffusion", where the generative process is designed such that larger updates to the sample are possible. The method is asymptotically exact and we demonstrate the effectiveness of our Decoupled Diffusion Sequential Monte Carlo (DDSMC) algorithm on both synthetic as well as protein and image data. Further, we demonstrate how the approach can be extended to discrete data.

Figures (17)

• Figure 1: Samples from DDSMC from the GMM experiments ($d_x=800$ and $d_y=1$) using either Tweedie's formula (three left-most figures) or the solution of the PF-ODE (three right-most figures) as a reconstruction, and using different values of $\eta$ in the generalized DAPS prior. Blue samples are from the posterior, while red samples are from DDSMC. More examples can be found in \ref{['app:gmm']}.
• Figure 2: Qualitative comparison between DDSMC (using PF-ODE as reconstruction and $\eta=0$) and other methods on the GMM experiments, $d_x=800$ and $d_y=1$. More qualitative comparisons can be found in \ref{['app:gmm']}
• Figure 3: Examples of generated images for the outpainting task. The DDSMC samples are ordered with $\eta=0$ to the left, $\eta=0.5$ in the middle, and $\eta=1$ to the right. Ground truth images are FFHQ images ID 100, 103, and 110 (see \ref{['tab:ffhq-attribution']} in the appendix for attribution).
• Figure 4: RMSD vs subsampling factor $k$ for the protein structure completion problem on the ATAD2 protein davison_mapping_2022 (PDB identifier 7qumTurberville2023-vs) and different noise levels $\sigma$: $0$ (left), $0.1$ (middle), $0.5$ (right). Comparing ADP-3D levy_solving_2024 and DDSMC with different number of particles.
• Figure 5: Qualitative results on binary MNIST using the discrete counterpart of DDSMC, D3SMC. Top is the ground truth, middle is measurement, and bottom is sampled image. More results can be found in \ref{['app:bmnist']}

Theorems & Definitions (6)

• Proposition 3.1
• Remark 3.2
• Proposition 1.1
• proof
• Remark 1.2
• Remark 1.3