度量-测度空间几何比较的DTM-签名
The DTM-signature for a geometric comparison of metric-measure spaces from samples
摘要：
在本文中，我们引入了DTM-签名的概念，R +上的测度，可以与任何度量测度空间。这个签名是基于Chazal、Cohen-Steiner和Mérigot提出的距离测度（DTM）。它导致度量测度空间之间的伪度量，上界为Gromov-Wasserstein距离。在一些几何假设下，我们推导出这个伪度量的下界。给定两个N-样本，我们还建立了一个渐近统计测试的基础上的DTM签名，拒绝的假设平等的两个基本的度量空间，保持测量等距。我们给这个测试强有力的理论依据，并提出了一个算法，其实施。
英文摘要：
In this paper, we introduce the notion of DTM-signature, a measure on R + that can be associated to any metric-measure space. This signature is based on the distance to a measure (DTM) introduced by Chazal, Cohen-Steiner and Mérigot. It leads to a pseudo-metric between metric-measure spaces, upper-bounded by the Gromov-Wasserstein distance. Under some geometric assumptions, we derive lower bounds for this pseudo-metric. Given two N-samples, we also build an asymptotic statistical test based on the DTM-signature, to reject the hypothesis of equality of the two underlying metric-measure spaces, up to a measure-preserving isometry. We give strong theoretical justifications for this test and propose an algorithm for its implementation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
