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Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

Eddie Aamari 1, 2, 3 Clément Levrard 4
1 SELECT - Model selection in statistical learning
LMO - Laboratoire de Mathématiques d'Orsay, Inria Saclay - Ile de France
2 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
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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Submitted on : Monday, November 20, 2017 - 2:16:59 AM
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Stability and Minimax Optimali...
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  • HAL Id : hal-01245479, version 3
  • ARXIV : 1512.02857


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



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