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This paper employs techniques from metric geometry and optimal transport theory to address questions related to the analysis of functional data on metric or metric-measure spaces, which we refer to as fields. Formally, fields are viewed as 1-Lipschitz mappings between Polish metric spaces with the domain possibly equipped with a Borel probability measure. We introduce field analogues of the Gromov-Hausdorff, Gromov-Prokhorov, and Gromov-Wasserstein distances, investigate their main properties and provide a characterization of the Gromov-Hausdorff distance in terms of isometric embeddings in a Urysohn universal field. Adapting the notion of distance matrices to fields, we formulate a discrete model, obtain an empirical estimation result that provides a theoretical basis for its use in functional data analysis, and prove a field analogue of Gromov's Reconstruction Theorem. We also investigate field versions of the Vietoris-Rips and neighborhood (or offset) filtrations and prove that they are stable with respect to appropriate metrics., Comment: Some missing references are added. An early version of this manuscript was posted to arXiv as arXiv:2208.04782

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., Comment: 23 pages