A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature
A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature
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DOI：
10.1214/19-ejs1539
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发表时间：
2019-01-01
影响因子：
1.1
通讯作者：
Brecheteau, Claire
中科院分区：
文献类型：
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作者：
Brecheteau, Claire
我们引入了 DTM 签名的概念，它是 R 上的一种度量，可以与任何度量-度量空间相关联。该签名基于 Chazal、Cohen-Steiner 和 Merigot 于 2009 年引入的函数测度距离 (DTM)。它导致度量测量空间之间的伪度量，其上方以 Gromov-Wasserstein 距离为界。该伪度量用于通过对两个 N 样本的观察来构建两个度量测量空间之间同构的统计测试。该测试基于子采样方法并具有理论保证。渐进地证明它具有正确的水平。此外，当 R-d 的紧凑子集支持这些度量时，会导出检验统计量分布及其子采样近似值之间的 L1-Wasserstein 距离的收敛率。这些速率取决于某些参数 rho > 1。此外，我们证明功率的界限为 exp(-C N-1/rho)，其中 C 与度量-测量空间之间的上述伪度量的平方成正比。在一些几何假设下，我们还推导了该伪度量的下界。提出了一种算法来实现该统计测试，并通过数值实验将其性能与其他方法的性能进行了比较。
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 function distance to a measure (DTM) introduced in 2009 by Chazal, Cohen-Steiner and Merigot. It leads to a pseudo-metric between metric-measure spaces, that is bounded above by the Gromov-Wasserstein distance. This pseudo-metric is used to build a statistical test of isomorphism between two metric-measure spaces, from the observation of two N-samples.The test is based on subsampling methods and comes with theoretical guarantees. It is proven to be of the correct level asymptotically. Also, when the measures are supported on compact subsets of R-d, rates of convergence are derived for the L1-Wasserstein distance between the distribution of the test statistic and its subsampling approximation. These rates depend on some parameter rho > 1. In addition, we prove that the power is bounded above by exp(-C N-1/rho), with C proportional to the square of the aforementioned pseudo-metric between the metric-measure spaces. Under some geometrical assumptions, we also derive lower bounds for this pseudometric.An algorithm is proposed for the implementation of this statistical test, and its performance is compared to the performance of other methods through numerical experiments.