Low Dimensional Sampling under Reconstructed Constraints

We study sampling in the presence of a compact lower-dimensional constraint manifold observed only through a representative point cloud. We first reconstruct the constraint using an adaptive local-convex-hull estimator and then target an ambient Gibbs law penalized by distance from the reconstructed set. The reconstruction need not be smooth or even be a manifold. Nevertheless, using only its Hausdorff accuracy, we prove a Wasserstein--$2$ approximation bound by working in the normal coordinates of the true manifold. A Gaussian random-walk Metropolis--Hastings algorithm on a fixed compact enclosure is uniformly geometrically ergodic for the penalized law. Combining reconstruction, stationary approximation, and mixing yields a finite-time guarantee for the complete procedure. With $N$ point-cloud observations, the stationary Wasserstein error is $O_{P}((\log N/N)^{1/d})$ for a $d$-dimensional constraint.