[Paper Review] Metric Graph Approximations of Geodesic Spaces

Facundo Mémoli, Osman Berat Okutan+1 more|arXiv (Cornell University)|Sep 14, 2018

Topological and Geometric Data Analysis14 references4 citations

TL;DR

This paper establishes that Reeb graphs of distance-to-point functions provide optimal metric graph approximations of compact geodesic spaces up to a multiplicative constant depending linearly on the first Betti number. By endowing Reeb graphs with an intrinsic metric derived from the distance function, the authors prove Gromov-Hausdorff stability and show that these Reeb metric graphs achieve approximation quality within a bounded factor of the best possible finite metric graph with a given Betti number bound.

ABSTRACT

We study the question of approximating a compact geodesic metric space by metric graphs satisfying a uniform upper bound on their first Betti number. We prove that, up to a suitable multiplicative constant, Reeb graphs of distance functions to a point provide optimal approximation in the Gromov-Hausdsorff sense.

Motivation & Objective

• To address the challenge of approximating compact geodesic spaces by finite metric graphs with bounded first Betti number.
• To resolve the issue that standard metric graph approximations can have high Betti numbers, making them topologically complex.
• To investigate whether Reeb graphs of 1-Lipschitz functions—especially distance-to-point functions—can serve as stable, low-complexity approximations.
• To establish the existence of a 'core' finite metric graph within Reeb metric graphs that preserves essential topological and geometric structure.
• To quantify the approximation quality of Reeb metric graphs and merge metric trees relative to the best possible finite metric graphs or trees.

Proposed method

• Define Reeb metric graphs as the quotient space of a geodesic space under the equivalence relation identifying points in the same connected component of sublevel sets of a 1-Lipschitz function, equipped with the intrinsic metric pulled back from the real line.
• Prove that Reeb metric graphs are 'tame' metric graphs: they contain a finite metric graph core isometrically embedded, with the remainder being a possibly infinite tree.
• Establish Gromov-Hausdorff stability of the Reeb metric graph construction under perturbations of the 1-Lipschitz function, with stability constants depending on the first Betti number and number of local minima.
• Introduce merge metric trees as a refinement of Reeb graphs, obtained by collapsing certain components, and prove their Gromov-Hausdorff stability.
• Use core theory in geodesic spaces to show that any finite metric graph core can be extended to include any given point, preserving the core property.
• Leverage homological and topological tools (e.g., long exact sequence in homology, Euler’s formula) to bound the number of edges in a minimal core, showing $E(G_0) \ leq 3b_1(G)$.

Experimental results

Research questions

• RQ1 Can Reeb graphs of 1-Lipschitz functions on compact geodesic spaces serve as low-complexity metric graph approximations with bounded first Betti number?
• RQ2 What is the Gromov-Hausdorff distance between a compact geodesic space and its Reeb metric graph, and how does it compare to the best possible finite metric graph approximation?
• RQ3 Is the Reeb metric graph construction stable under small perturbations of the input function?
• RQ4 Can merge metric trees—derived from Reeb graphs—provide optimal tree-like approximations of geodesic spaces?
• RQ5 How does the approximation error of Reeb metric graphs scale with the first Betti number and geometric properties like area in 2-dimensional spaces?

Key findings

• Reeb metric graphs of distance-to-point functions are Gromov-Hausdorff stable, with the stability constant depending linearly on the first Betti number of the space and the number of local minima of the function.
• The approximation error $\rho_p(X) = d_{\mathrm{GH}}(X, \mathrm{R}_f)$ satisfies $\rho_p(X) \leq C(n,X) \cdot \delta_n(X)$, where $C(n,X)$ is linear in $n$ and $b_1(X)$, showing that Reeb graphs are optimal up to a multiplicative constant.
• For compact geodesic spaces with Hausdorff dimension 2, the approximation error $\rho_p(X)$ is bounded above in terms of the area of the space.
• Merge metric trees induced by distance-to-point functions achieve optimal tree approximation: $d_{\mathrm{GH}}(X, T_f) \leq 13 \cdot \delta_0(X)$, meaning they are optimal up to a universal multiplicative constant.
• Reeb metric graphs are 'tame' in the sense that they contain a finite metric graph core isometrically embedded, with the complement being a tree, ensuring topological and geometric control.
• Any finite metric graph core in a compact geodesic space can be extended to include any given point, preserving the core property and enabling inductive constructions.