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Distributions of angles in random packing on spheres

Author(s): Cai, T; Fan, J; Jiang, T

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Abstract: This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in ℝ p 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. © 2013 Tony Cai, Jianqing Fan and Tiefeng Jiang.
Publication Date: 1-Jun-2013
Citation: Cai, T, Fan, J, Jiang, T. (2013). Distributions of angles in random packing on spheres. Journal of Machine Learning Research, 14 (1837 - 1864
ISSN: 1532-4435
EISSN: 1533-7928
Pages: 1837 - 1864
Type of Material: Journal Article
Journal/Proceeding Title: Journal of Machine Learning Research

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