# Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

1 SELECT - Model selection in statistical learning Inria Saclay - Ile de France, LMO - Laboratoire de Mathématiques d-Orsay, CNRS - Centre National de la Recherche Scientifique : UMR 2 DATASHAPE - Understanding the Shape of Data CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France 3 LM-Orsay - Laboratoire de Mathématiques d-Orsay 4 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Abstract : In this paper we consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X} n = \left\{X 1,\ldots,X n ight\}\subset \mathbb{R}^D$ 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, we construct an estimator $\hat{M}$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. $\hat{M}$ 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 with reach condition. 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.

Keywords : Manifold reconstruction Minimax optimality Tangential Delaunay com- plexes Denoising Tangent space estimation

Author: Eddie Aamari - Clément Levrard -

Source: https://hal.archives-ouvertes.fr/