Tweedie Moment Projected Diffusions for Inverse Problems

Diffusion generative models unlock new possibilities for inverse problems as they allow for the incorporation of strong empirical priors into the process of scientific inference. Recently, diffusion models received significant attention for solving inverse problems by posterior sampling, but many challenges remain open due to the intractability of this sampling process. Prior work resorted to Gaussian approximations to conditional densities of the reverse process, leveraging Tweedie's formula to parameterise its mean, complemented with various heuristics. In this work, we leverage higher order information using Tweedie's formula and obtain a finer approximation with a principled covariance estimate. This novel approximation removes any time-dependent step-size hyperparameters required by earlier methods, and enables higher quality approximations of the posterior density which results in better samples. Specifically, we tackle noisy linear inverse problems and obtain a novel approximation to the gradient of the likelihood. We then plug this gradient estimate into various diffusion models and show that this method is optimal for a Gaussian data distribution. We illustrate the empirical effectiveness of our approach for general linear inverse problems on toy synthetic examples as well as image restoration using pretrained diffusion models as the prior. We show that our method improves the sample quality by providing statistically principled approximations to diffusion posterior sampling problem.