Ensemble Kalman Sampling and Diffusion Prior in Tandem: A Split Gibbs Framework

We study the inverse problems in the derivative-free setting, where the forward model only permits black-box access. Traditional derivative-free methods, such as Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), have limited applicability in this problem due to their reliance on the prior density access (up to a normalizing constant), which is intractable in many applications. Recent works leveraging diffusion models (DMs) as flexible priors show promise for high-dimensional inverse problems, but we find they often deviate from the true posterior, even in linear Gaussian case. To address these limitations, we propose SG-EKDP (Split Gibbs with Ensemble Kalman sampling and Diffusion Prior), a novel algorithm that integrates ensemble Kalman sampling and diffusion prior within a split Gibbs sampling framework. Our method provably converges to the posterior in the mean-field limit for general linear inverse problems and remains effective in nonlinear settings via local linearization. We evaluate the effectiveness of SG-EKDP across various inverse problems. Numerical experiments on linear Gaussian and Gaussian mixture show that SG-EKDP can accurately approximate the true posterior. It also achieves strong empirical performance on high-dimensional image restoration tasks including both linear and nonlinear problems.