Image: Random almost orthogonal sets

Description: Lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1,1]n as a function of dimension, n. A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within π/2 ±0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ±0.037π/2 then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each n, 20 pairwise almost orthogonal chains where constructed numerically. Distribution of the length of these chains is presented. Boxplots show the second and third quartiles of this data for each n, red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound, N≤eϵ2n4[−ln⁡(1−θ)]12{\displaystyle N\leq e^{\frac {\epsilon ^{2}n}{4}}[-\ln(1-\theta )]^{\frac {1}{2}}}, and green curve shows a refined estimate.[1]
Title: Random almost orthogonal sets
Credit: Own work
Author: Agor153
Usage Terms: Creative Commons Attribution 3.0
License: CC BY 3.0
License Link: http://creativecommons.org/licenses/by/3.0
Attribution Required?: Yes