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 ($\mathrm{CGW}$) 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 $\mathrm{CGW}$ as a shape metric for chiral objects.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2608.27774 [math.OC]

(or arXiv:2608.27774v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2608.27774 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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