提出可高效计算的内积型格罗莫夫-沃瑟斯坦距离，适合异构数据对齐。
Sliced Inner Product Gromov-Wasserstein Distances
- 基于切片技术设计内积型格罗莫夫-沃瑟斯坦距离，保持旋转不变性
- 理论证明该方法在高维下具有良好计算与统计性质
- 适用于文本聚类和语言模型表征对比等实际任务
格罗莫夫-沃瑟斯坦（GW）问题提供了一种通过匹配内在几何结构来对齐异构数据集的框架，但其在高维问题中的统计与计算可扩展性仍是挑战。切片技术为可扩展性提供了有吸引力的路径，然而与沃瑟斯坦距离不同，通常无法在单维情况下得到闭式解。本文解决了内积代价（IGW）情形下的这一难题，提出了具有自然旋转不变性的切片内积型格罗莫夫-沃瑟斯坦距离，并系统研究了其结构与计算特性。数值实验验证了理论结果，并展示了在文本数据异构聚类和语言模型表征比较中的应用。
原文摘要 · Abstract (English)
The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.
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