# Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction Article - Février 2018

Eddie Aamari, Clément Levrard

Eddie Aamari, Clément Levrard, « Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction  », Discrete and Computational Geometry, février 2018. ISSN 0179-5376

Abstract

In this paper we consider the problem of optimality in manifold reconstruction. A random sample $\mathbbX_n = \left{X_1,\ldots,X_n\right}\subset \mathbbR^D$ 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 $\hatM$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. $\hatM$ 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.