Aaron R. Voelker, Jan Gosmann, and Terrence C. Stewart. Efficiently sampling vectors and coordinates from the n-sphere and n-ball. Technical Report, Centre for Theoretical Neuroscience, Waterloo, ON, 01 2017. URL: https://www.researchgate.net/publication/312056739_Efficiently_sampling_vectors_and_coordinates_from_the_n-sphere_and_n-ball, doi:10.13140/RG.2.2.15829.01767/1.
@techreport{voelker2017,
title = {Efficiently sampling vectors and coordinates from the n-sphere and n-ball},
year = {2017},
month = {01},
institution = {Centre for Theoretical Neuroscience},
address = {Waterloo, ON},
abstract = {We provide a short proof that the uniform distribution of points for the n-ball is equivalent to the uniform distribution of points for the (n + 1)-sphere projected onto n dimensions. This implies the surprising result that one may uniformly sample the n-ball by instead uniformly sampling the (n + 1)-sphere and then arbitrarily discarding two coordinates. Consequently, any procedure for sampling coordinates from the uniform (n + 1)-sphere may be used to sample coordinates from the uniform n-ball without any modification. For purposes of the Semantic Pointer Architecture (SPA), these insights yield an efficient and novel procedure for sampling the dot-product of vectors—sampled from the uniform ball—with unit-length encoding vectors.},
issn = {CTN-TR-20170104-01},
author = {Aaron R. Voelker and Jan Gosmann and Terrence C. Stewart},
pdf = {/files/publications/voelker.2017.pdf},
url = {https://www.researchgate.net/publication/312056739_Efficiently_sampling_vectors_and_coordinates_from_the_n-sphere_and_n-ball},
doi = {10.13140/RG.2.2.15829.01767/1}
}