[Paper Review] Gromov-Hausdorff Approximation of Metric Spaces with Linear Structure

Frédéric Chazal, Jian Sun|arXiv (Cornell University)|May 6, 2013

Topological and Geometric Data Analysis26 references4 citations

TL;DR

This paper proposes a method to reconstruct filamentary metric structures—modeled as metric graphs—from discrete metric space data using Gromov-Hausdorff approximation. It introduces the α-Reeb graph, a computationally efficient variant of the Reeb graph, and proves that the resulting metric graph approximates the original space within a bounded distortion, with theoretical guarantees depending on the Betti number and edge length distribution of the target graph.

ABSTRACT

In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few synthetic and real data sets.

Motivation & Objective

• To address the metric reconstruction problem for data sampled around 1-dimensional filamentary structures, such as road networks or geological faults.
• To provide a provably accurate approximation of such structures using metric graphs under the Gromov-Hausdorff distance.
• To overcome the limitations of standard Reeb graphs in discrete, non-geodesic data by introducing the α-Reeb graph with computational and theoretical advantages.
• To enable efficient, almost linear-time reconstruction of metric graphs from point cloud data without requiring Euclidean embeddings.
• To ensure that the reconstructed graph preserves metric structure and directional information, especially in applications like GPS trace analysis.

Proposed method

• The method constructs a neighborhood graph from discrete data points using local distances, then computes a shortest-path metric to approximate a geodesic space.
• It introduces the α-Reeb graph, a variant of the Reeb graph derived from the distance function to a root point on the neighborhood graph, which is more amenable to efficient computation.
• The α-Reeb graph is computed via the Mapper algorithm, which allows for scalable and robust topological abstraction of the data.
• Theoretical guarantees are established by bounding the Gromov-Hausdorff distance between the input metric space and the reconstructed α-Reeb graph.
• The algorithm uses a 1-Lipschitz quotient map that preserves distances to the root point and contracts other distances, minimizing metric distortion.
• The method handles directional data by stacking consecutive GPS points into higher-dimensional vectors, preserving traffic flow and intersection structure.

Experimental results

Research questions

• RQ1 Can a discrete metric space sampled from a filamentary structure be accurately reconstructed as a metric graph under the Gromov-Hausdorff distance?
• RQ2 How can the Reeb graph be adapted to enable efficient and stable computation on discrete, non-geodesic data?
• RQ3 What is the theoretical bound on the Gromov-Hausdorff error between the input space and the reconstructed metric graph?
• RQ4 Can the algorithm preserve directional and topological features such as road network crossings from GPS traces?
• RQ5 How does the choice of root point affect the metric distortion in the reconstruction?

Key findings

• The Gromov-Hausdorff distance between the input space $X$ and the reconstructed α-Reeb graph $G$ is bounded by $d_{GH}(X,G) < 2(eta_1(G)+1)(17 + 8N_{E,G'}(8ar{\varepsilon}))\varepsilon$, where $N_{E,G'}(8\varepsilon)$ is the number of short edges in the target graph.
• If the target metric graph has all edges longer than $8\varepsilon$, the bound simplifies to $d_{GH}(X,G) < 34(\beta_1(G)+1)\varepsilon$, providing a clean approximation guarantee.
• The algorithm runs in almost linear time, making it scalable for large-scale geometric data such as GPS traces.
• The method successfully reconstructs a highway crossing from synthetic and real GPS data, preserving the non-crossing topology of roads via directional stacking in higher dimensions.
• The reconstructed α-Reeb graph accurately captures the topology and metric structure of the underlying filamentary network, as demonstrated on both synthetic and real-world datasets.
• Theoretical analysis shows that the first Betti number of the underlying metric graph can be inferred from the data using persistence homology, suggesting potential for topological consistency.