Statistics Papers
Document Type
Journal Article
Date of this Version
7-2013
Publication Source
Journal of Machine Learning Research
Volume
14
Start Page
1837
Last Page
1864
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.
Keywords
random angle, uniform distribution on sphere, empirical law, maximum of random variables, minimum of random variables, extreme-value distribution, packing on sphere
Recommended Citation
Cai, T., Fan, J., & Jiang, T. (2013). Distributions of Angles in Random Packing on Spheres. Journal of Machine Learning Research, 14 1837-1864. Retrieved from https://repository.upenn.edu/statistics_papers/136
Date Posted: 27 November 2017
This document has been peer reviewed.