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 μn. Nevertheless, in many situations the data points are not located on the geometric shape but in the neighborhood of it, and μn 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.
"Deconvolution for the Wasserstein metric and geometric inference." Electron. J. Statist. 5 1394 - 1423, 2011. https://doi.org/10.1214/11-EJS646