Abstract
This paper is a short presentation of recent results about Wasserstein deconvolution for topological inference published in [1]. A distance function to measures has been defined in [2] 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 for he Wasserstein distance W 2. 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.
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Caillerie, C., Chazal, F., Dedecker, J., Michel, B. (2013). Deconvolution for the Wasserstein Metric and Geometric Inference. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_62
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DOI: https://doi.org/10.1007/978-3-642-40020-9_62
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