论文标题
通用映射和几何域功能数据的分析
Universal Mappings and Analysis of Functional Data on Geometric Domains
论文作者
论文摘要
本文解决了在研究数据中出现的功能度量几何形状的问题，例如在几何域上记录的信号或加权网络节点上的信号。包含此类物体的数据集出现在许多科学和实际兴趣领域。例如，$ f $可以代表功能性磁共振图像，也可以代表带有属性或偏好标记的社交网络的节点，其中基础度量结构由最短路径距离，通勤距离或扩散距离给出。正式地，这些可能被视为在公制空间上定义的函数，有时配备了其他结构，例如概率度量，在这种情况下，域称为度量量化空间，或者仅简单地称为$ mm $ $ - $ -SPACE。我们的主要目标是三倍：（i）制定指标，使我们能够建模并量化功能数据的变化，可能具有不同的域； （ii）研究这些指标的原则经验估计； （iii）构建一个通用函数，``包含''的所有功能，其域和范围是抛光的（可分离且完整的度量）空间，假设Lipschitz的规律性。后者在很大程度上符合为结构数据（度量空间）构建通用空间的精神，其调查可以追溯到20世纪初，并且对公制几何学具有经典的兴趣。
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.
