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Distributions of Angles in Random Packing on Spheres
Tony Cai1, Jianqing Fan2, Tiefeng Jiang3
1Statistics Department The Wharton School University of Pennsylvania Philadelphia, PA 19104, USA TCAI@WHARTON.UPENN.EDU.
Journal of Machine Learning Research : JMLR
|October 18, 2014
Summary
As the number of random unit vectors increases, their pairwise angles approach a distribution, confirming high-dimensional vectors are nearly orthogonal. This study analyzes these asymptotic behaviors in fixed and growing dimensions.
Area of Science:
- Probability Theory
- High-Dimensional Geometry
- Random Matrix Theory
Background:
- Understanding the geometric configurations of random vectors is crucial in various scientific fields.
- The behavior of random vectors in high dimensions presents unique and often counterintuitive properties.
Purpose of the Study:
- To investigate the asymptotic behaviors of pairwise angles between n randomly distributed unit vectors.
- To analyze these behaviors in both fixed and growing dimensional settings as n approaches infinity.
Main Methods:
- Derivation of limiting empirical distributions for random angles.
- Analysis of limiting distributions for extreme angles.
- Mathematical modeling of vector distributions in Euclidean space (ℝ^p).
Main Results:
- Established the limiting empirical distribution of pairwise angles for large n.
- Determined the limiting distributions of extreme angles in both fixed and growing dimensions.
- Highlighted distinct behaviors between fixed and growing dimensional cases.
Conclusions:
- The findings precisely characterize the phenomenon of near-orthogonality among high-dimensional random vectors.
- Results have implications for statistics, machine learning, and open problems in physics and mathematics.
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