Computing the Gromov--Hausdorff distance using gradient methods

TL;DR

This work introduces its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance and uses it to obtain a new bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric.

Abstract

The Gromov--Hausdorff distance measures the difference in shape between metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance. The optimality guarantee is enabled by the fact that the search space of our approach is not constrained to a generalization of bijections, unlike in other relaxations such as the Gromov--Wasserstein distance. We suggest conditional gradient descent for solving the relaxation in cubic time per iteration, and demonstrate its performance on metric spaces of hundreds of points. In particular, we use it to obtain a new bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric. Our approach is implemented as a Python package dGH.

Figures (3)

• Figure 1: An example with $|X| = |Y| = 4$ for which the mapping pairs minimizing the distortion are not bijective. The dotted lines represent one of such mapping pairs. The distances in $X$ and $Y$ take only two distinct values, whose ratio exceeds 2.
• Figure 2: Performance of dgh on synthetic metric spaces with $n=200$ points.
• Figure 3: The relation between $S^1_\varepsilon$ and $H^2_\varepsilon$ induced by a mapping pair recovered by dgh.

Theorems & Definitions (20)

• Theorem 1
• Corollary : to Theorem \ref{['thm:c_threshold']}
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 4
• proof