Deconvolution for the Wasserstein Metric and Geometric Inference

Abstract : Recently, \cite{Chazaletal11} 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 $\nu$, if $\nu$ is close enough to a measure $\mu$ concentrated on this shape. Here, close enough means that the Wasserstein distance $W_2$ between $\mu$ and $\nu$ is sufficiently small. Given a point cloud, a natural candidate for $\nu$ is the empirical measure $\mu_n$. Nevertheless, in many situations the data points are not located on the geometric shape but in the neighborhood of it, and $\mu_n$ can be too far from $\mu$. 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.
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Submitted on : Monday, July 11, 2011 - 2:00:55 PM
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Claire Caillerie, Frédéric Chazal, Jérôme Dedecker, Bertrand Michel. Deconvolution for the Wasserstein Metric and Geometric Inference. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2011, 5, pp.1394-1423. ⟨10.1214/11-EJS646⟩. ⟨inria-00607806⟩

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