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

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Date Posted: 27 November 2017

This document has been peer reviewed.