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On Metrics for Analysis of Functional Data on Geometric Domains

Soheil Anbouhi, Washington Mio, Osman Berat Okutan

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.

On Metrics for Analysis of Functional Data on Geometric Domains

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.
Paper Structure (14 sections, 34 theorems, 128 equations, 2 figures)

This paper contains 14 sections, 34 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main Results
  4. Organization
  5. Gromov-Hausdorff Distance for Metric Fields
  6. The Urysohn Field
  7. Gromov-Prokhorov Distance for Metric-Measure Fields
  8. Gromov-Wasserstein Distance for Metric-Measure Fields
  9. Gromov-Wasserstein Through Functional Curvature
  10. Topological Multifiltrations and Their Stability
  11. Neighborhood Multifiltrations
  12. Vietoris-Rips Multifiltrations
  13. Summary and Discussion
  14. Appendix

Key Result

Lemma 2.4

If $\mathscr{Y},\mathscr{Z}_1,\mathscr{Z}_2$ are $B$-fields and $\Phi \colon \mathscr{Y} \hookrightarrow \mathscr{Z}_1$, $\Psi \colon \mathscr{Y} \hookrightarrow \mathscr{Z}_2$ are isometric embeddings, then $d_Z \colon Z \times Z \to [0,\infty)$ is a well-defined metric on $Z$ and $\pi_Z \colon Z \

Figures (2)

  • 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 $t=0.99$.
  • 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 $t=0.1$.

Theorems & Definitions (87)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: Gluing Lemma
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6: Metric Field Distortion
  • Definition 2.7
  • Lemma 2.8
  • ...and 77 more