This paper addresses problems in functional metric geometry that arise in the study of data such as signals recorded on geometric domains or on the nodes of weighted networks. Datasets comprising such objects arise in many domains of scientific and practical interest. For example, $f$ could represent a functional magnetic resonance image, or the nodes of a social network labeled with attributes or preferences, where the underlying metric structure is given by the shortest path distance, commute distance, or diffusion distance. Formally, these may be viewed as functions defined on metric spaces, sometimes equipped with additional structure such as a probability measure, in which case the domain is referred to as a metric-measure space, or simply $mm$-space. Our primary goal is threefold: (i) to develop metrics that allow us to model and quantify variation in functional data, possibly with distinct domains; (ii) to investigate principled empirical estimations of these metrics; (iii) to construct a universal function that ``contains'' all functions whose domains and ranges are Polish (separable and complete metric) spaces, assuming Lipschitz regularity. The latter is much in the spirit of constructing universal spaces for structural data (metric spaces) whose investigation dates back to the early 20th century and are of classical interest in metric geometry.

中文翻译：

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几何域上功能数据的通用映射和分析

本文解决了在研究数据（例如记录在几何域或加权网络节点上的信号）中出现的功能度量几何问题。包含此类对象的数据集出现在许多具有科学和实际意义的领域。例如，$f$ 可以表示功能性磁共振图像，或标有属性或偏好的社交网络的节点，其中基础度量结构由最短路径距离、通勤距离或扩散距离给出。形式上，这些可以被视为在度量空间上定义的函数，有时配备了诸如概率度量之类的附加结构，在这种情况下，该域被称为度量-度量空间，或简称为 $mm$-space。我们的主要目标有三个：(i) 制定指标，使我们能够对功能数据的变化进行建模和量化，可能具有不同的领域；(ii) 研究这些指标的原则性经验估计；(iii) 假设 Lipschitz 正则性，构造一个“包含”所有域和范围是波兰（可分离和完全度量）空间的函数的通用函数。后者在很大程度上本着为结构数据（度量空间）构建通用空间的精神，其研究可以追溯到 20 世纪初，并且对度量几何具有经典的兴趣。假设 Lipschitz 正则性，所有域和范围都是波兰（可分离和完整度量）空间的函数。后者在很大程度上本着为结构数据（度量空间）构建通用空间的精神，其研究可以追溯到 20 世纪初，并且对度量几何具有经典的兴趣。假设 Lipschitz 正则性，所有域和范围都是波兰（可分离和完整度量）空间的函数。后者在很大程度上本着为结构数据（度量空间）构建通用空间的精神，其研究可以追溯到 20 世纪初，并且对度量几何具有经典的兴趣。