Empirical Convergence of Even-Order Gromov-Wasserstein Functionals

We study the sample complexity of empirical plug-in estimation for the powered even-order Gromov-Wasserstein functional between compactly supported probability measures on ${\mathbb{R}}^{d_x}$ and ${\mathbb{R}}^{d_y}$. For every fixed pair of integers $r,k\ge 1$, we prove that the two-sample empirical error is bounded at the rate $n^{-2\mathrm{/}\max \{\min \{d_x,d_y\},4\}}$, up to a logarithmic factor in the critical case $\min \{d_x,d_y\}=4$. This extends the known quadratic Euclidean upper rate to the full powered even-order family. The proof uses a polynomial decomposition of the even-order GW functional, a generalized duality formula reducing the coupling-dependent term to a compact family of ordinary optimal transport problems, and entropy estimates for semiconcave dual potentials.