Efficiently sampling vectors and coordinates from the n-sphere and n-ball

Tech Report, 2017

Aaron R. Voelker, Jan Gosmann, Terrence C. Stewart

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.

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CTN Tech Report

Issn
CTN-TR-20170104-01
Address
Waterloo, ON
Month
01
Doi
10.13140/RG.2.2.15829.01767/1
Institution
Centre for Theoretical Neuroscience

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