Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

In this paper we consider the problem of optimality in manifold reconstruction. A random sample Xn = {X1, . . . , Xn} ⊂ R composed of points lying on a d-dimensional submanifold M , with or without outliers drawn in the ambient space, is observed. Based on the Tangential Delaunay Complex [4], we construct an estimator M̂ that is ambient isotopic and Hausdorffclose to M with high probability. The estimator M̂ is built from existing algorithms. In a model without outliers, we show that this estimator is asymptotically minimax optimal for the Hausdorff distance over a class of submanifolds defined with 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 denoising procedure and a tangent space estimator both based on local principal component analysis (PCA) are studied.