Fitting a Putative Manifold to Noisy Data

Abstract

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