Beyond Mean Curvature: Pathwise Langevin Sampling on Learned Data Manifolds

Underdetermined inverse problems admit multiple plausible solutions, yet learned priors entangle data-manifold geometry with training-density preferences, hindering controlled exploration. We propose geometric-prior exploration: the pretrained diffusion model defines the manifold of plausible solutions, while task potential and inverse temperature control density relative to manifold volume. To sample this target when the manifold is known only through a pretrained diffusion model, we introduce $\mathrm{II}L$, a manifold Langevin discretization combining soft landing with a pathwise curvature correction based on the second fundamental form while avoiding explicit computation of the mean-curvature vector. We construct the required directional-curvature estimator directly from diffusion scores without forming a tangent basis or the full curvature tensor. Under inexact geometric access, we establish stationary weak-bias guarantees and show that repeated learned projection can amplify geometric error as the step size decreases. Our pathwise correction further improves the deviation of the stationary distribution from the manifold from $O(\sqrt{\Delta t})$ under the standard mean-curvature correction to $O(\Delta t)$, where $\Delta t$ is the discretization step size. Experiments on synthetic manifolds, image inpainting, human-pose editing, and image inverse problems validate the predicted numerical behavior, demonstrate temperature-controlled exploration of task-compatible solutions, and improve reconstruction refinement at comparable runtime.