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 manifoldM, and corrupted by a small amount of gaussian noise. How can we produce a manifoldMo whose Hausdorff distance to M is small and whose reach is not much smaller than the reach of M?