Convergence of empirical Gromov-Wasserstein distance

We study rates of convergence for estimation of the Gromov-Wasserstein (GW) 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 $n^{-\frac{2}{\min\{d_x,d_y\}}}$ in $L^1$ 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 GW distance and Wasserstein Procrustes. Furthermore, we establish matching minimax lower bounds (up to logarithmic factors) for estimating the GW distance. Finally, we establish deviation inequalities for the error of empirical GW in cases where two marginals have compact supports, exponential tails, or finite polynomial moments. The deviation inequalities yield that the same rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ holds for empirical GW also with high probability.