蔡天文 Tony Cai 2014-10-10 6:16 PM

Distributions of Angles in Random Packing on Spheres

Abstract

This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in ℜp 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.


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ABOUT THE AUTHOR

蔡天文 Tony Cai

宾夕法尼亚大学沃顿商学院Dorothy Silberberg 统计学讲席教授、应用数学及计算科学教授

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