Empirical Convergence of Even-Order Gromov-Wasserstein Functionals
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910213608570880 |
|---|---|
| author | Paliy, Vasyl |
| author_facet | Paliy, Vasyl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11108 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Empirical Convergence of Even-Order Gromov-Wasserstein Functionals Paliy, Vasyl Probability Statistics Theory 60B10 (Primary), 62G05, 49Q22, 60F05 (Secondary) 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. |
| title | Empirical Convergence of Even-Order Gromov-Wasserstein Functionals |
| topic | Probability Statistics Theory 60B10 (Primary), 62G05, 49Q22, 60F05 (Secondary) |
| url | https://arxiv.org/abs/2605.11108 |