Deconvolution for the Wasserstein Metric and Geometric Inference
Deconvolution for the Wasserstein Metric and Geometric Inference
复制标题
Wasserstein 度量和几何推理的反卷积
DOI：
10.1007/978-3-642-40020-9_62
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发表时间：
2011
影响因子：
6.8
通讯作者：
B. Michel
中科院分区：
文献类型：
--
作者：
Claire Caillerie;F. Chazal;J. Dedecker;B. Michel
本文是文[1]中关于拓扑推理的Wasserstein反卷积的最新结果的一个简短介绍。在[2]中定义了一个距离函数来回答概率环境中的几何推理问题。根据他们的结果，当测度μ与测度ν足够接近时，只要Wasserstein距离W2，就可以用到测度ν的距离来恢复形状的拓扑性质.给定一个点云，ν的一个自然候选是经验测度μ n。然而，在许多情况下，数据点并不位于几何形状上，而是位于它的邻域中，并且μ n可能离μ太远。在反卷积框架中，我们考虑对经典的核反卷积估计量进行轻微的修改，并给出了该估计量的一致性结果和收敛速度。一些模拟实验说明了反卷积方法及其在各种形状和各种噪声分布的几何推断中的应用。
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.