超越普罗克拉斯提斯距离：一种捕捉手性的多重线性格罗莫夫-瓦瑟斯坦距离
Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality
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中文总结 AI 辅助
该研究针对现有形状分析指标无法区分形状与其镜像的问题，提出了捕捉手性的多重线性格罗莫夫-瓦瑟斯坦距离，开发了高效计算算法并通过实验验证了其有效性。
中文摘要 AI 辅助
高效且稳健地分析形状数据在众多科学学科中至关重要。手性是众多应用（尤其是分子科学）中的基本属性，但现有形状分析指标无法区分形状与其镜像。为解决这一差距，我们引入了格罗莫夫-瓦瑟斯坦目标的多重线性泛化。在温和假设下，该目标会在由对称群G商化的概率分布所表示的形状之间产生距离。特别地，对于G=SO(d)，我们引入了对於手性敏感的手性格罗莫夫-瓦瑟斯坦（CGW）距离。我们确立了多重线性格罗莫夫-瓦瑟斯坦距离的稳健性属性，并开发了高效算法来计算它们，通过将耦合投影到低维空间来重构底层优化问题。我们推导了局部和近似全局解的算法，为这些问题提供了完全多项式时间近似方案。我们通过数值实验验证了该框架，证明了CGW作为手性物体形状指标的有效性。
英文摘要
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.
