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
2 DATASHAPE - Understanding the Shape of Data
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay
4 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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.

Mots clés: Manifold reconstruction; Denoising; Tangent space estimation; Minimax optimality; Tangential Delaunay Complexes; Decluttering

Domaines: Statistiques [math.ST]

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