  We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M}
\subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$.
There are many algorithms (e.g., Isomap) that are used in practice to fit
manifolds and thus reduce the dimensionality of a given data set. Ideally, the
estimate $\mathcal{M}_\mathrm{put}$ of $\mathcal{M}$ should be an actual
manifold of a certain smoothness; furthermore, $\mathcal{M}_\mathrm{put}$
should be arbitrarily close to $\mathcal{M}$ in Hausdorff distance given a
large enough sample. Generally speaking, existing manifold learning algorithms
do not meet these criteria. Fefferman, Mitter, and Narayanan (2016) have
developed an algorithm whose output is provably a manifold. The key idea is to
define an approximate squared-distance function (asdf) to $\mathcal{M}$. Then,
$\mathcal{M}_\mathrm{put}$ is given by the set of points where the gradient of
the asdf is orthogonal to the subspace spanned by the largest $n - d$
eigenvectors of the Hessian of the asdf. As long as the asdf meets certain
regularity conditions, $\mathcal{M}_\mathrm{put}$ is a manifold that is
arbitrarily close in Hausdorff distance to $\mathcal{M}$. In this paper, we
define two asdfs that can be calculated from the data and show that they meet
the required regularity conditions. The first asdf is based on kernel density
estimation, and the second is based on estimation of tangent spaces using local
principal components analysis.
