Posterior Transport Matching for Inverse Problems

Posterior sampling must preserve uncertainty while remaining consistent with the observation. We introduce Posterior Transport Matching (PTM), a training-free framework that pairs numerical transport maps with stochastic input updates, termed rematching. The key is to supply each map with an input distribution that it sends to the posterior. We characterize the minimum-energy isotropic diffusion realizing these compatible laws and identify its objective with path-space relative entropy. Exact probability-flow maps admit Gaussian rematching. For computation, entropy projection yields an objective for the Gaussian transition mean, combining an expected potential with a squared-distance penalty. Denoised observation energy approximates the potential; an analytic approximation requires only the observation operator. A single bound on output KL separates potential approximation from finite computation. Controlled experiments show how the appropriate transition depends on the map. Across eight FFHQ inverse problems, PTM achieves the highest reported PSNR across eight FFHQ inverse problems and favorable quality–cost tradeoffs, with further evaluation on ImageNet.