Distributions of angles in random packing on spheres

Authors:
Tony Cai;Jianqing Fan;Tiefeng Jiang
Affiliations:
Statistics Department, The Wharton School, University of Pennsylvania, Philadelphia, PA;Department of Operation Research and Financial Engineering, Princeton University, Princeton, NJ;School of Statistics, University of Minnesota, Minneapolis, MN
Venue:
The Journal of Machine Learning Research
Year:
2013

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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.