Distributions of angles in random packing on spheres
Distributions of angles in random packing on spheres
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DOI：
10.5555/2567709.2567722
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发表时间：
2013-06
期刊：
影响因子：
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通讯作者：
Tony Cai;Jianqing Fan;Tiefeng Jiang
中科院分区：
文献类型：
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作者：
Tony Cai;Jianqing Fan;Tiefeng Jiang
本文研究了[公式：见正文]中n个随机均匀分布的单位向量的两两夹角在维数p固定或随n增长时，当点数n → ∞时的渐近性质.对于这两种设置，我们推导出的随机角度的极限经验分布和极限分布的极端角度。结果揭示了有趣的差异，在这两种设置，并提供了一个精确的民间传说，“所有的高维随机向量几乎总是几乎相互正交”的表征。还讨论了统计和机器学习的应用以及与物理和数学中一些开放问题的联系。
This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in [Formula: see text] as the number of points n → ∞, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that "all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and machine learning and connections with some open problems in physics and mathematics are also discussed.