Gromov-Wasserstein Barycenter Surrogates: Statistical Methodology, Distributional Limits and Applications
Abstract
We introduce statistical theory for the matching of finitely many objects, represented as metric measure spaces (mm-spaces). The approach is based on the second lower bound (SLB) of the Gromov-Wasserstein distance and thus is able to identify deviations in the distributions of the (pairwise) distances within each mm-space. We introduce a surrogate of the SLB barycenter which can be easily computed and expressed explicitly in terms of the distance distributions of each object. When comparing m mm-spaces for n randomly drawn samples in each space, the resulting statistic then can be calculated efficiently in O(m · n2 log(n)) basic operations. We derive the asymptotic distribution and finite-sample bounds of the proposed test statistic, which serves as a basis for a variety of tools for statistical inference, specifically an asymptotic test for pose-invariant object discrimination and a classification method (based on the SLB barycenter) with controlled error rates. These methods are investigated in simulations and applied to the structural comparison of protein domains.
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BibTeX
@misc{steinkamp2026gromovwasserstein,
title = {{Gromov-Wasserstein Barycenter Surrogates: Statistical Methodology, Distributional Limits and Applications}},
author = {Florian Steinkamp and Luis-Alberto Rodríguez and Jan Victor Otte and Axel Munk},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11670},
eprint = {2609.11670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11670},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gromov-Wasserstein Barycenter Surrogates: Statistical Methodology, Distributional Limits and Applications
AU - Florian Steinkamp
AU - Luis-Alberto Rodríguez
AU - Jan Victor Otte
AU - Axel Munk
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11670
UR - https://arxiv.org/abs/2609.11670
AB - We introduce statistical theory for the matching of finitely many objects, represented as metric measure spaces (mm-spaces). The approach is based on the second lower bound (SLB) of the Gromov-Wasserstein distance and thus is able to identify deviations in the distributions of the (pairwise) distances within each mm-space. We introduce a surrogate of the SLB barycenter which can be easily computed and expressed explicitly in terms of the distance distributions of each object. When comparing m mm-spaces for n randomly drawn samples in each space, the resulting statistic then can be calculated efficiently in O(m · n^2 log(n)) basic operations. We derive the asymptotic distribution and finite-sample bounds of the proposed test statistic, which serves as a basis for a variety of tools for statistical inference, specifically an asymptotic test for pose-invariant object discrimination and a classification method (based on the SLB barycenter) with controlled error rates. These methods are investigated in simulations and applied to the structural comparison of protein domains.
ER -