## 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.

Original language | English (US) |
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Pages (from-to) | 1837-1864 |

Number of pages | 28 |

Journal | Journal of Machine Learning Research |

Volume | 14 |

State | Published - Jun 2013 |

## All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence

## Keywords

- Empirical law
- Extreme-value distribution
- Maximum of random variables
- Minimum of random variables
- Packing on sphere
- Random angle
- Uniform distribution on sphere