Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality

2026-08-27

AI Summary

本文提出了一种新的多线性Gromov-Wasserstein距离，以解决现有形状分析方法无法区分物体与其镜像的问题，特别是引入了对映性的CGW距离，并开发了高效算法。

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.

Problem

• chirality
• shape analysis
• Gromov-Wasserstein distance

Innovation

• multilinear Gromov-Wasserstein
• chirality
• shape analysis
• Chiral Gromov-Wasserstein (CGW)
• efficient algorithms