Fitting a putative manifold to noisy data

Abstract: 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 compact manifold M, and is corrupted by a small amount of i.i.d gaussian noise. How can we produce a manifold M whose Hausdorff distance to M is small and whose reach (normal injectivity radius) is not much smaller than the reach of M?
This is joint work with Charles Fefferman, Sergei Ivanov, Yaroslav Kurylev, and Matti Lassas.