  We consider the problem of optimality in manifold reconstruction. A random
sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$
composed of points close to a $d$-dimensional submanifold $M$, with or without
outliers drawn in the ambient space, is observed. Based on the Tangential
Delaunay Complex, we construct an estimator $\hat{M}$ that is ambient isotopic
and Hausdorff-close to $M$ with high probability. The estimator $\hat{M}$ is
built from existing algorithms. In a model with additive noise of small
amplitude, we show that this estimator is asymptotically minimax optimal for
the Hausdorff distance over a class of submanifolds satisfying a reach
constraint. Therefore, even with no a priori information on the tangent spaces
of $M$, our estimator based on Tangential Delaunay Complexes is optimal. This
shows that the optimal rate of convergence can be achieved through existing
algorithms. A similar result is also derived in a model with outliers. A
geometric interpolation result is derived, showing that the Tangential Delaunay
Complex is stable with respect to noise and perturbations of the tangent
spaces. In the process, a decluttering procedure and a tangent space estimator
both based on local principal component analysis (PCA) are studied.
