标题： 经验Gromov-Wasserstein距离的收敛性 显示英文标题

标题： Convergence of empirical Gromov-Wasserstein distance

摘要： 我们研究了Gromov-Wasserstein距离估计的收敛速度。对于分别支持在$\R^{d_x}$和$\R^{d_y}$的紧子集上的两个边缘分布，具有$\min \{ d_x,d_y \} > 4$，先前的工作基于$n$个独立同分布样本建立了插件经验估计量的速率$n^{-\frac{2}{\min\{d_x,d_y\}}}$。我们将这一基本结果扩展到具有无界支撑的边缘分布，仅假设有限的多项式矩。我们用于上界证明的技术可以调整以获得惩罚Wasserstein对齐的样本复杂度结果，该结果涵盖了无界情况下的Gromov-Wasserstein距离和Wasserstein Procrustes。此外，我们建立了估计Gromov-Wasserstein距离的匹配最小最大下界（忽略对数因子）。 显示英文摘要

摘要： We study rates of convergence for estimation of the Gromov-Wasserstein distance. For two marginals supported on compact subsets of $\R^{d_x}$ and $\R^{d_y}$, respectively, with $\min \{ d_x,d_y \} > 4$, prior work established the rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ for the plug-in empirical estimator based on $n$ i.i.d. samples. We extend this fundamental result to marginals with unbounded supports, assuming only finite polynomial moments. Our proof techniques for the upper bounds can be adapted to obtain sample complexity results for penalized Wasserstein alignment that encompasses the Gromov-Wasserstein distance and Wasserstein Procrustes in unbounded settings. Furthermore, we establish matching minimax lower bounds (up to logarithmic factors) for estimating the Gromov-Wasserstein distance.