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Stability and Approximations for Decorated Reeb Spaces

Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, Florian Russold

Abstract

Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

Stability and Approximations for Decorated Reeb Spaces

Abstract

Given a map from a topological space to a metric space , a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of . In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.
Paper Structure (16 sections, 13 theorems, 15 equations, 8 figures, 1 algorithm)

This paper contains 16 sections, 13 theorems, 15 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Decorated Reeb Spaces
  3. Reeb Spaces and Smoothings
  4. Reeb Radius
  5. Metrizing the Reeb Space
  6. Decorations
  7. Gromov-Hausdorff Stability
  8. Metric Fields and Multiscale Comparisons
  9. Stability of Filtration Decorated Reeb Spaces
  10. Stability of Barcode Decorated Reeb Spaces
  11. Decorated Reeb Spaces for Combinatorial Graphs
  12. An Algorithm for the Reeb Radius Function
  13. Finite Approximations of Decorated Reeb Spaces
  14. Simplifying Graphs by Smoothing
  15. Computational Examples
  16. ...and 1 more sections

Key Result

Theorem 2

For $(X,f) \in \mathcal{F}_{M}$ and $x, y \in X$, $\pi_f^\epsilon(x)=\pi_f^\epsilon(y)$ if and only if $f(x)=f(y)$ and $\rho_f(x,y) \leq \epsilon$.

Figures (8)

  • Figure 1: The Decorated Reeb Graph pipeline. (a) A point cloud, noisily sampled from a space homotopy equivalent to $(S^1 \times S^1) \vee S^1$. (b) The (estimated) Reeb graph associated to the point cloud, with respect to the height function. (c) To each node of the Reeb graph, we associate a filtered space, which is a sequence of subsets of the original point cloud. (d) Taking persistent homology of the filtered space associated to each node yields a decorated Reeb graph; each node is attributed with a persistence diagram.
  • Figure 2: The graph of a generic function $f:[0,1]\to\mathbb{R}$ is its own Reeb graph. Notice that the Reeb radius is not symmetric in $x$ and $y$.
  • Figure 3: On the left is a graph endowed with an $\mathbb{R}$-valued function $g$ given by node height. The other two graphs are the spanning trees generated by Algorithm \ref{['alg:reeb_radius']} with respect to the roote node $x$, with a directed edge $[w,v]$ for each $v,w$ that is processed inside the conditional statement. At each node, the value of $\rho_g(x,v)$ is written. The directed path from $v$ to $x$ realizes $\rho_g(x,v)$.
  • Figure 4: A graph with function $g$ given by node height, its Reeb graph and two smoothings.
  • Figure 5: The USAir97 graph of airports and connections, endowed with the PageRank function, together with a smoothed Reeb graph illustrating hub structure.
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 1: Reeb radius
  • Theorem 2
  • Lemma 3
  • Proposition 4
  • Remark 5
  • Definition 6
  • Definition 7: Filtration decoration for Reeb Spaces
  • Definition 8: Barcode decoration for Reeb Spaces
  • Remark 9
  • Definition 10: (r,s)-Correspondences between metric fields
  • ...and 22 more