Gromov-Wasserstein重心替代：统计方法、分布极限及应用
Gromov-Wasserstein Barycenter Surrogates: Statistical Methodology, Distributional Limits and Applications
- University of Göttingen（哥廷根大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于Gromov-Wasserstein距离第二下界的重心替代统计方法，用于度量测度空间的匹配与分类，推导渐近分布并应用于蛋白质结构域比较。
AI中文摘要：
我们为有限多个对象的匹配引入了统计理论，这些对象以度量测度空间（mm-空间）表示。该方法基于Gromov-Wasserstein距离的第二下界（SLB），因此能够识别每个mm-空间内（成对）距离分布的偏差。我们引入了SLB重心的一个替代量，该替代量易于计算，并可用每个对象的距离分布显式表达。当比较m个mm-空间，每个空间随机抽取n个样本时，所得统计量可在O(m·n²log(n))次基本运算内高效计算。我们推导了所提检验统计量的渐近分布和有限样本界，这为多种统计推断工具提供了基础，具体包括用于姿态不变对象区分的渐近检验和一种基于SLB重心的分类方法，该方法具有受控的错误率。这些方法在模拟中进行了研究，并应用于蛋白质结构域的结构比较。
英文摘要：
We introduce statistical theory for the matching of finitely many objects, represented as metric measure spaces (mm-spaces). The approach is based on the second lower bound (SLB) of the Gromov-Wasserstein distance and thus is able to identify deviations in the distributions of the (pairwise) distances within each mm-space. We introduce a surrogate of the SLB barycenter which can be easily computed and expressed explicitly in terms of the distance distributions of each object. When comparing $m$ mm-spaces for $n$ randomly drawn samples in each space, the resulting statistic then can be calculated efficiently in $O(m \cdot n^2 \log(n))$ basic operations. We derive the asymptotic distribution and finite-sample bounds of the proposed test statistic, which serves as a basis for a variety of tools for statistical inference, specifically an asymptotic test for pose-invariant object discrimination and a classification method (based on the SLB barycenter) with controlled error rates. These methods are investigated in simulations and applied to the structural comparison of protein domains.
