Deconvolution for the Wasserstein metric and geometric inference

Abstract

Recently, Chazal, Cohen-Steiner and Mérigot have defined a distance function to measures to answer geometric inference problems in a probabilistic setting. According to their result, the topological properties of a shape can be recovered by using the distance to a known measure ν, if ν is close enough to a measure μ concentrated on this shape. Here, close enough means that the Wasserstein distance W₂ between μ and ν is sufficiently small. Given a point cloud, a natural candidate for ν is the empirical measure μ_n. Nevertheless, in many situations the data points are not located on the geometric shape but in the neighborhood of it, and μ_n can be too far from μ. In a deconvolution framework, we consider a slight modification of the classical kernel deconvolution estimator, and we give a consistency result and rates of convergence for this estimator. Some simulated experiments illustrate the deconvolution method and its application to geometric inference on various shapes and with various noise distributions.