Computing the Gromov–Hausdorff distance using first-order methods
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The Gromov–Hausdorff distance measures the difference in shape between compact 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 the Frank–Wolfe algorithm for solving the relaxation in O(n^3) time per iteration, and numerically 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.
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