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arXiv · 2605.11108

Empirical Convergence of Even-Order Gromov-Wasserstein Functionals

Vasyl Paliy
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Abstract

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 $r,k\geq 1$, we prove that the two-sample empirical error is bounded at the rate $n^{-2/\max\{\min\{d_x,d_y\},4\}}$, 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.

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Vasyl Paliy. 2026-05-11. Empirical Convergence of Even-Order Gromov-Wasserstein Functionals. https://arxiv.org/abs/2605.11108

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