On Metrics for Analysis of Functional Data on Geometric Domains

TL;DR

This paper develops metric geometry tools for analyzing functional data on geometric domains, introducing new distances and models, and establishing theoretical properties and stability results.

Contribution

It introduces field analogues of key Gromov distances, formulates a discrete model, and proves a Gromov-type reconstruction theorem for fields.

Findings

Defined field Gromov-Hausdorff, Prokhorov, Wasserstein distances

Provided an empirical estimation framework for functional data analysis

Proved stability of field Vietoris-Rips and offset filtrations

Abstract

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.