We find the minimax rate of convergence in Hausdorff distance for estimating
a manifold M of dimension d embedded in R^D given a noisy sample from the
manifold. We assume that the manifold satisfies a smoothness condition and that
the noise distribution has compact support. We show that the optimal rate of
convergence is n^{-2/(2+d)}. Thus, the minimax rate depends only on the
dimension of the manifold, not on the dimension of the space in which M is
embedded.

We consider the problem of reliably finding filaments in point clouds.
Realistic data sets often have numerous filaments of various sizes and shapes.
Statistical techniques exist for finding one (or a few) filaments but these
methods do not handle noisy data sets with many filaments. Other methods can be
found in the astronomy literature but they do not have rigorous statistical
guarantees. We propose the following method. Starting at each data point we
construct the steepest ascent path along a kernel density estimator.