Empirical Convergence of Even-Order Gromov-Wasserstein Functionals

TL;DR

This paper analyzes the sample complexity of empirical estimation for even-order Gromov-Wasserstein functionals, establishing convergence rates that extend previous results to the full powered even-order family.

Contribution

It provides the first convergence rate bounds for empirical plug-in estimators of powered even-order Gromov-Wasserstein functionals, generalizing known Euclidean results.

Findings

Sample error bounds at rate n^{-2/ max{min{d_x,d_y},4}}

Extends quadratic Euclidean upper rate to all powered even-order GW functionals

Uses polynomial decomposition and duality to analyze the functional's convergence

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.