Computing the Gromov--Hausdorff distance using gradient methods

TL;DR

This paper introduces a quadratic relaxation method for computing the Gromov--Hausdorff distance between metric spaces, using gradient methods to efficiently approximate the measure of shape difference.

Contribution

It presents a novel convex relaxation approach and a gradient-based optimization algorithm for the Gromov--Hausdorff distance, with theoretical guarantees and practical implementation.

Findings

Achieved efficient computation on metric spaces with hundreds of points.

Provided a new bound for the Gromov--Hausdorff distance between the circle and hemisphere.

Implemented the method as the Python package dGH.

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.