Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality
Abstract
Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group G. In particular, for G = SO(d), we introduce the Chiral Gromov-Wasserstein (CGW) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of CGW as a shape metric for chiral objects.
Related
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BibTeX
@misc{soubrier2026beyond,
title = {{Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality}},
author = {Clément Soubrier and Geoffrey Woollard and Andrew Warren and Khanh Dao Duc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27774},
eprint = {2608.27774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27774},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality
AU - Clément Soubrier
AU - Geoffrey Woollard
AU - Andrew Warren
AU - Khanh Dao Duc
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.27774
UR - https://arxiv.org/abs/2608.27774
AB - Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group G. In particular, for G = SO(d), we introduce the Chiral Gromov-Wasserstein (CGW) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of CGW as a shape metric for chiral objects.
ER -