超越 Procrustes 距离：一种捕捉手性的多线性 Gromov-Wasserstein 距离

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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.

Comments:	57 pages, 6 figures
Subjects:	Optimization and Control (math.OC); Machine Learning (cs.LG)
MSC classes:	49Q22 (Primary) 90C26, 65K10 (Secondary)
Cite as:	arXiv:2608.27774 [math.OC]
	(or arXiv:2608.27774v2 [math.OC] for this version)
	https://doi.org/10.48550/arXiv.2608.27774 arXiv-issued DOI via DataCite

Submission history

From: Clément Soubrier [view email]
[v1] Thu, 27 Aug 2026 23:14:38 UTC (6,485 KB)
[v2] Mon, 5 Oct 2026 20:10:08 UTC (6,485 KB)