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We consider the problem of optimality in manifold reconstruction. A randomsample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$composed of points close to a $d$-dimensional submanifold $M$, with or withoutoutliers drawn in the ambient space, is observed. Based on the TangentialDelaunay Complex, we construct an estimator $\hat{M}$ that is ambient isotopicand Hausdorff-close to $M$ with high probability. The estimator $\hat{M}$ isbuilt from existing algorithms. In a model with additive noise of smallamplitude, we show that this estimator is asymptotically minimax optimal forthe Hausdorff distance over a class of submanifolds satisfying a reachconstraint. Therefore, even with no a priori information on the tangent spacesof $M$, our estimator based on Tangential Delaunay Complexes is optimal. Thisshows that the optimal rate of convergence can be achieved through existingalgorithms. A similar result is also derived in a model with outliers. Ageometric interpolation result is derived, showing that the Tangential DelaunayComplex is stable with respect to noise and perturbations of the tangentspaces. In the process, a decluttering procedure and a tangent space estimatorboth based on local principal component analysis (PCA) are studied.

Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction