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

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.

If using an AI tool for generating an image, this tool is likely to be based on something called a diffusion model. A diffusion model generates an image (or some other kind of data) in steps, starting from something that is complete noise, and gradually makes this into an image.In this work, we develop a method that can guide such model so that it generates something based (or conditioned) on some ”partial” data, also known as an inverse problem. For example, if you have an image where some parts are missing, and you have a diffusion model which can generate images, a method like the one developed in this work would enabling using this model to “fill in” the missing parts. Our method is based on a method from statistics called sequential Monte Carlo. The developed method is a way of bridging modern and powerful AI methods with classical statistical methods, reaping the benefits from both.