Fitting a Putative Manifold to Noisy Data.

In the present work, we give a solution to the following question from manifold learning. Suppose data belonging to a high dimensional Euclidean space is drawn independently, identically distributed from a measure supported on a low dimensional twice differentiable embedded manifold, and corrupted by a small amount of gaussian noise. How can we produce a manifold whose Hausdorff distance to the true manifold is small and whose reach is not much smaller than the reach of the true manifold?