[Paper Review] Lectures on Hausdorff and Gromov-Hausdorff Distance Geometry

Alexey Avgustinovich Tuzhilin|arXiv (Cornell University)|Dec 1, 2020

Topological and Geometric Data Analysis7 references4 citations

TL;DR

This paper provides a comprehensive graduate-level introduction to Hausdorff and Gromov–Hausdorff distance geometry, establishing foundational topological and metric space theory before developing the theory of Gromov–Hausdorff convergence and its applications. The key contribution is a novel characterization of clique covering and chromatic numbers in finite graphs using Gromov–Hausdorff distances to metric simplexes, enabling algorithmic computation of these graph invariants via geometric distance metrics.

ABSTRACT

The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff distance for separable spaces in terms of their isometric images in \ell_\infty, correspondences, Gromov-Hausdorff distance in terms of correspondences. (6) Epsilon-isometries and Gromov-Hausdorff distance. (7) Irreducible correspondences and Gromov-Hausdorff distance. (8) Gromov-Hausdorff convergence, inheritance of metric and topological properties while Gromov-Hausdorff convergence. (9) Gromov-Hausdorff space (GH-space), optimal correspondences, existence of closed optimal correspondences for compact metric spaces, GH-space is geodesic. (10) Cover number, packing number, total boundedness, completeness, and separability of GH-space. (11) mst-spectrum in terms of GH-distances to simplexes, Steiner problem in GH-space. (12) GH-distance to simplexes with more points, GH-distance to simplexes with at most the same number of points. (13) Generalized Borsuk problem, solution of Generalized Borsuk problem in terms of GH-distances, clique covering number and chromatic number of simple graphs, their dualities, calculating these numbers in terms of GH-distances.

Motivation & Objective

• To provide a rigorous, self-contained introduction to general topology, metric spaces, and curve theory as a foundation for distance geometry.
• To develop the theory of Hausdorff and Gromov–Hausdorff distances in compact and complete metric spaces.
• To establish the geodesic and completeness properties of the Gromov–Hausdorff space of compact metric spaces.
• To apply Gromov–Hausdorff distance to solve classical problems in graph theory, particularly clique covering and chromatic number computation.

Proposed method

• Utilizes standard topological constructions (subspaces, product, quotient, Vietoris topology) and metric space embeddings to define and analyze convergence and compactness.
• Introduces the Hausdorff metric on hyperspaces of compact subsets and proves limits theorems for complete and compact metric spaces.
• Defines the Gromov–Hausdorff distance via irreducible correspondences and proves existence of optimal correspondences between compact metric spaces.
• Applies the Arzelà–Ascoli theorem and Hopf–Rinow condition to establish existence of shortest curves and geodesics in metric spaces.
• Constructs a metric on finite graphs by assigning distance $a$ to adjacent vertices and $b > a$ to non-adjacent ones, linking graph cliques to subsets of smaller diameter.
• Uses Theorem 8.7 to relate the Gromov–Hausdorff distance $2d_{GH}(a\Delta_k, V)$ to the minimal number of subsets of diameter less than $\operatorname{diam}V$ needed to cover $V$.

Experimental results

Research questions

• RQ1 How can the Gromov–Hausdorff distance be used to characterize the minimal number of subsets of strictly smaller diameter needed to cover a finite metric space?
• RQ2 What is the relationship between the Gromov–Hausdorff distance from a finite metric space to a simplex and the clique covering number of an associated graph?
• RQ3 Can the chromatic number of a graph be computed via Gromov–Hausdorff distances to metric simplexes?
• RQ4 Under what conditions does the Gromov–Hausdorff space of compact metric spaces become geodesic or totally bounded?
• RQ5 How does the $\operatorname{mst}$-spectrum of a finite metric space relate to its Gromov–Hausdorff distance to simplexes?

Key findings

• The clique covering number $\theta(G)$ of a finite simple graph $G$ equals $m+1$, where $m$ is the largest integer $k$ such that $2d_{GH}(a\Delta_k, V) = b$, with $a < b \leq 2a$.
• For a finite graph $G$ with metric defined by $a$ for adjacent vertices and $b > a$ for non-adjacent ones, $\theta(G) = m+1$ where $m$ is the maximal $k$ with $2d_{GH}(a\Delta_k, V) = b$.
• The chromatic number $\gamma(G)$ of a graph $G$ equals $m+1$, where $m$ is the largest $k$ such that $2d_{GH}(a\Delta_k, V) = b$ when distances are reversed: $b$ for adjacent, $a$ for non-adjacent vertices.
• The Gromov–Hausdorff space of compact metric spaces is complete and separable, and it is geodesic, meaning any two compact metric spaces can be joined by a minimizing geodesic.
• The $\operatorname{mst}$-spectrum of a finite metric space can be computed using Gromov–Hausdorff distances to simplexes, linking combinatorial optimization to metric geometry.
• If a finite metric space $X$ has diameter $d$, then $X$ cannot be partitioned into $m$ subsets of strictly smaller diameter if and only if $2d_{GH}(\lambda\Delta_m, X) = d$ for $\lambda = d/2$.