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.

翻译：Gromov-Hausdorff距离度量了度量空间之间的形状差异，并构成了组合优化中一个公认的难题。我们提出了该距离在一个凸多胞体上的二次松弛，其解可保证得到Gromov-Hausdorff距离。最优性保证得益于我们方法的搜索空间不局限于双射的推广，这与Gromov-Wasserstein距离等其他松弛方法不同。我们建议采用条件梯度下降法来求解该松弛，每次迭代的时间复杂度为立方阶，并在包含数百个点的度量空间上展示了其性能。特别地，我们利用该方法获得了配备欧几里得度量的单位圆与单位半球面之间Gromov-Hausdorff距离的一个新界。我们的方法已实现为Python包dGH。