The paper shows a practical trick to improve a class of diffusion-based posterior samplers. Instead of running a particle sampler from the very noisy starting point, the authors propose to begin at an intermediate time. They draw samples from an easier, “softened” inverse problem in the clean data space, then map those samples exactly back to the noisy intermediate target using a closed-form Gaussian bridge. Only the remaining particle-based steps are run after that.

Concretely, the work targets samplers that introduce observations by Gaussian “tilts” of diffusion model marginals. The authors observe these tilted targets can be pulled back to a clean-space posterior that is less tightly conditioned on the data. They sample that softened clean posterior approximately with moment-matching posterior sampling (MMPS). Because the conditional from clean to noisy space is Gaussian and known, the approximate clean samples are transported exactly to the diffusion-time target through an analytic Gaussian bridge. The remainder of the sequential Monte Carlo (SMC) sampler MCGDiff is then run from that intermediate time onward.

The idea matters because it improves finite-particle performance where particle methods often fail. On a structured Gaussian-mixture inverse problem the hybrid (MMPS initialization plus the MCGDiff suffix) roughly halves the sliced Wasserstein distance (a measure of distributional error) compared with MCGDiff alone at the same particle count. When the posterior-relevant mode is rare under the prior, the improvement exceeds an order of magnitude. The paper evaluates three Gaussian-mixture benchmarks where the exact posterior and the intermediate targets can be sampled analytically. Performance is reported against exact posterior samples using sliced Wasserstein distance, and the authors use an effective resampling rule (resample only when effective sample size drops below 0.7 of the particle count).