Latent Annealed Posterior Sampling with a Variance-Preserving Bridge for Sparsely Observed Random Fields

Jiankun Lei, Tian Qi

Posterior inference for random fields from sparse observations presents two major challenges: accurately characterizing complex spatial uncertainty and preventing conditional sampling costs from growing substantially as the spatial grid is refined. To address these challenges, we propose latent annealed posterior sampling (LAPS). LAPS first maps the random field into a latent space with reduced dimensionality, while a coordinate decoder composed of Branch and Trunk networks enables field evaluation at arbitrary spatial locations. Flow matching is then used to learn the latent prior distribution. Based on this learned prior, posterior sampling is performed entirely in the latent space, with the observation likelihood incorporated progressively through an annealing procedure. To control local covariance inflation, we introduce a variance-preserving (VP) bridge that allocates a covariance budget between the transported latent state and a Gaussian perturbation. Numerical experiments demonstrate that LAPS accurately approximates posterior distributions, preserves distinct posterior modes, and reduces the dependence of conditional sampling costs on spatial resolution. Across two…