AI 中文总结
该研究分析弱信号 regime 下球形 K-均值的初始化偏差，通过 von Mises-Fisher 混合模型推导误差缩放规律，证明其可保留初始化结构，为相关算法优化提供理论依据。
AI 中文摘要
我们针对弱信息方向混合场景研究球形 K-均值的初始化偏差问题。我们将观测数据建模为 K 分量 von Mises-Fisher 混合模型，其浓度参数 κ 很小，对应高离散 regime，此时数据对潜在方向的信息提供有限。我们的分析从极限情况 κ=0（对应球面上的均匀分布）入手，此时单总体球形 K-均值更新完全由初始化模板诱导的 Voronoi 镶嵌决定。在固定维度 d 中，对于均匀随机初始化，当 K→∞ 时更新后的模板渐近与初始值对齐：平均平方测地误差缩放为 O(K^{-2/(d-1)})，最坏情况误差为 O((log K/K)^{2/(d-1)})。随后我们证明，在小正值 κ 的弱信号 regime 下，总体更新仍是该极限映射的 O(κ) 扰动。因此，在弱信号 regime 中，球形 K-均值可保留初始化诱导的结构，尽管存在真实但高度离散的方向信号。
英文摘要
We study initialization bias in spherical $K$-means for weakly informative directional mixtures. We model the observations by a $K$-component von Mises-Fisher mixture with a small concentration parameter $κ$, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case $κ=0$ (corresponding to a uniform distribution over the sphere), where one population spherical $K$-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension $d$, the updated templates become asymptotically aligned with their initial values as $K\to\infty$: the average squared geodesic error scales as $O(K^{-2/(d-1)})$, while the worst-case error is $O((\log K/K)^{2/(d-1)})$. We then show that, in the weak-signal regime of small positive $κ$, the population update remains an $O(κ)$ perturbation of this limiting map. Thus, in the weak-signal regime, spherical $K$-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.
