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