提出新形状度量方法，能区分分子等物体的左右手镜像。
Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality
- 基于多线性优化构建新型距离度量，可捕捉形状的旋向性。
- 在分子数据集上验证有效，能正确区分互为镜像的结构。
- 算法高效，支持快速近似求解，适合大规模形状分析。
高效稳健地分析形状数据在多个科学领域至关重要。尽管旋向性在诸多应用中是基本属性（尤其在分子科学中），现有形状分析度量无法区分形状与其镜像。为此，我们引入了Gromov-Wasserstein目标的多线性推广。在温和假设下，该目标能生成形状间的距离，将形状表示为关于对称群 $G$ 商化后的概率分布。特别地，当 $G = SO(d)$ 时，我们提出旋向性Gromov-Wasserstein（CGW）距离，对旋向性敏感。我们建立了多线性Gromov-Wasserstein距离的鲁棒性，并开发了高效计算算法，通过将耦合投影到低维空间重构底层优化问题。推导出局部与近似全局解的算法，实现完全多项式时间近似方案。数值实验验证了框架的有效性，表明CGW作为旋向性物体形状度量具有显著效果。
原文摘要 · Abstract (English)
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.
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