格罗莫夫-瓦瑟斯坦量化与聚类：结构、速率与算法
Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms
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- University of Tübingen（蒂宾根大学）
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中文总结 AI 辅助
本文研究格罗莫夫-瓦瑟斯坦量化与聚类，证明其解的存在性，推导量化速率，提出类k-均值算法，实验显示其可用于3D形状、神经网络剪枝等场景，近似质量符合理论速率。
中文摘要 AI 辅助
聚类是一类基础的数据分析技术，最重要的代表是k-均值这类基于质心的方法。这类方法与量化问题密切相关，量化问题旨在用离散概率测度近似一般概率测度。例如，k-均值对应瓦瑟斯坦距离下的量化。瓦瑟斯坦量化在固定空间内对点进行聚类，而本文研究格罗莫夫-瓦瑟斯坦（Gromov-Wasserstein，GW）量化，其额外目标是对空间的环境几何进行聚类。我们证明了GW量化问题解的存在性，并给出一种刻画，为k-均值算法（Lloyd算法）的类似算法提供依据，以对解进行数值近似。我们进一步计算了GW语境中常用欧氏几何的量化速率，并将其与标准瓦瑟斯坦量化速率关联起来。最后，数值实验表明，GW量化为超越常规聚类方法的建模需求提供了诸多可能（例如针对3D形状的测地线距离或神经网络的结构化剪枝），且所提出的算法能产生有用的数值解，其近似质量通常与理论最优速率相符。
英文摘要
Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example, $k$-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the $k$-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.
