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Journal Articles Discrete and Computational Geometry Year : 2018

Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

Abstract

In this paper we consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ 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.
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Dates and versions

hal-01245479 , version 1 (17-12-2015)
hal-01245479 , version 2 (20-06-2016)
hal-01245479 , version 3 (20-11-2017)

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Eddie Aamari, Clément Levrard. Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction. Discrete and Computational Geometry, 2018. ⟨hal-01245479v3⟩
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