Distance distributions and inverse problems for metric measure spaces
Distance distributions and inverse problems for metric measure spaces
复制标题
度量测度空间的距离分布和反问题
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Tom Needham
中科院分区：
文献类型：
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作者：
F. Mémoli;Tom Needham
数据科学、形状分析和对象分类中的应用经常需要比较不同环境空间中定义的概率分布。要做到这一点，需要一个距离的概念，在一个给定的度量测度空间，也就是说，紧凑的度量空间赋予概率措施。这样的距离通常被定义为度量测量空间不变量之间的比较，例如距离分布（在文献中也称为形状分布、距离直方图或形状上下文）。一般来说，用距离分布定义的距离实际上是伪度量，因为它们在比较非同构空间时可能会消失。本文的目标是建立一个正式的框架来评估距离分布的判别力，即这些伪度量未能定义适当的度量的程度。我们制定了几个精确的反问题，这些不变量和回答他们在几个类别的度量测度空间，包括类别的平面曲线，我们给一个反例的曲线直方图猜想的Brinkman和Olver，类别的嵌入和黎曼流形，我们获得球刚性的结果，和类别的度量图，其中，我们得到了局部内射性的结果沿着线的经典工作的Boutin和Kemper的点云配置。反问题的进一步contextualized的Gromov-Wasserstein距离的度量测度空间的空间，这是由最初的蒙赫制定的最佳运输的启发的一个变种的介绍。
Applications in data science, shape analysis, and object classification frequently require comparison of probability distributions defined on different ambient spaces. To accomplish this, one requires a notion of distance on a given class of metric measure spaces—that is, compact metric spaces endowed with probability measures. Such distances are typically defined as comparisons between metric measure space invariants, such as distance distributions (also referred to as shape distributions, distance histograms, or shape contexts in the literature). Generally, distances defined in terms of distance distributions are actually pseudometrics, in that they may vanish when comparing nonisomorphic spaces. The goal of this paper is to set up a formal framework for assessing the discrimininative power of distance distributions, that is, the extent to which these pseudometrics fail to define proper metrics. We formulate several precise inverse problems in terms of these invariants and answer them in several categories of metric measure spaces, including the category of plane curves, where we give a counterexample to the curve histogram conjecture of Brinkman and Olver, the categories of embedded and Riemannian manifolds, where we obtain sphere rigidity results, and the category of metric graphs, where we obtain a local injectivity result along the lines of classical work of Boutin and Kemper on point cloud configurations. The inverse problems are further contextualized by the introduction of a variant of the Gromov–Wasserstein distance on the space of metric measure spaces, which is inspired by the original Monge formulation of optimal transport.
影响因子：
0.5
作者：
Balch, Brenden;Peterson, Chris;Shonkwiler, Clayton
通讯作者：
Shonkwiler, Clayton