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 with $O(n^3)$-time iterations for solving the relaxation and numerically demonstrate its performance on metric spaces of hundreds of points. In particular, we obtain a new upper 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距离衡量紧度量空间之间的形状差异，是组合优化中一个著名的困难问题。我们引入其在凸多胞体上的二次松弛，其解可证明地给出Gromov–Hausdorff距离。这一最优性保证得益于：与Gromov–Wasserstein距离等其他松弛方法不同，我们的搜索空间并不局限于双射的泛化。我们提出采用每轮迭代时间复杂度为$O(n^3)$的Frank–Wolfe算法求解该松弛，并在包含数百个点的度量空间上数值验证其性能。特别地，我们获得了单位圆与单位半球（均配备欧几里得度量）之间Gromov–Hausdorff距离的新上界。我们的方法已实现为Python包dGH。