On metrics for analysis of functional data on geometric domains

Soheil Anbouhi; Washington Mio; Osman Berat Okutan

doi:10.3934/fods.2024046

Article Contents

2025, Volume 7, Issue 3: 671-704. Doi: 10.3934/fods.2024046

This issue Previous Article Next Article Transport map unadjusted Langevin algorithms: Learning and discretizing perturbed samplers

On metrics for analysis of functional data on geometric domains

1.
Department of Mathematics and Computer Science, Western Carolina University, USA
2.
Department of Mathematics, Florida Sate University, USA
3.
Max Planck Institute for Mathematics in the Sciences, Germany

^*Corresponding author: Soheil Anbouhi
^*Corresponding author: Soheil Anbouhi

Received: September 2023

Revised: May 2024

Early access: October 21, 2024

Published online: October 21, 2024

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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.

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Mathematics Subject Classification: Primary: 51F30, 60B05, 60B10.

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Figure 1. Neighborhoods associated with a scalar field defined on a finite set of points $ X $ sampled from two circles: (a) $ N^{r}(X,E) $; (b) $ N^{r,s}(X, \mathcal{E}) $; (c) $ N^{r,s,t}( \mathcal{E}) $. The parameter values are $ r = 0.8 $, $ s = 0.1 $ and $ t = 0.99 $

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Figure 2. Simplicial complexes associated with a scalar field defined on a weighted finite set of points $ X $: (a) the $ V\!R $-complex $ V\!R^r (X) $; (b) the $ m $-field $ V\!R $-complex $ V\!R^{r,s}( \mathcal{X}) $; the $ mm $-field $ V\!R $-complex $ V\!R^{r,s,t}( \mathcal{X}) $. The parameter values are $ r = 1.5 $, $ s = 1 $ and $ t = 0.1 $

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