  A common observation in data-driven applications is that high dimensional
data has a low intrinsic dimension, at least locally. In this work, we consider
the problem of estimating a $d$ dimensional sub-manifold of $\mathbb{R}^D$ from
a finite set of noisy samples. Assuming that the data was sampled uniformly
from a tubular neighborhood of $\mathcal{M}\in \mathcal{C}^k$, a compact
manifold without boundary, we present an algorithm that takes a point $r$ from
the tubular neighborhood and outputs $\hat p_n\in \mathbb{R}^D$, and
$\widehat{T_{\hat p_n}\mathcal{M}}$ an element in the Grassmanian $Gr(d, D)$.
We prove that as the number of samples $n\to\infty$ the point $\hat p_n$
converges to $p\in \mathcal{M}$ and $\widehat{T_{\hat p_n}\mathcal{M}}$
converges to $T_p\mathcal{M}$ (the tangent space at that point) with high
probability. Furthermore, we show that the estimation yields asymptotic rates
of convergence of $n^{-\frac{k}{2k + d}}$ for the point estimation and
$n^{-\frac{k-1}{2k + d}}$ for the estimation of the tangent space. These rates
are known to be optimal for the case of function estimation.
