A minimax-optimal estimator for hypersurfaces

H. González-Vázquez, B. Pateiro-López, A. Rodríguez-Casal

Many high-dimensional Euclidean data tend to lie near a low-dimensional manifold, an assumption known as the manifold hypothesis. Manifold estimation techniques aim to estimate the underlying manifold to understand the true structure of the data. In this work, we propose a manifold estimator that builds upon the Euclidean Distance Transform (EDT) estimator proposed by Genovese et al. (2012) and improves it by making use of a shape restriction known as r-convexity. More specifically, we propose to use the r-convex hull of the sample, which generalizes the usual convex hull, to estimate the support of the sampling distribution. Our estimator achieves the optimal minimax rate of convergence in Hausdorff distance, up to logarithmic factor, to estimate manifolds with dimension one unit less than that of the ambient space, known as hypersurfaces. We also illustrate the estimator with a real data application.

Keywords: Manifold estimation, Minimax rate, r-convex hull, Set estimation

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