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From Sean Owen <sro...@myrrix.com>
Subject [math] Improving speed of UnitSphereRandomVectorGenerator
Date Wed, 30 Jan 2013 11:54:50 GMT
Hello all, I have a small proposal to improve the speed of
UnitSphereRandomVectorGenerator. This class picks a random point on
the unit n-sphere -- a unit vector, chosen uniformly from all possible
directions.

It does so using a rejection process -- generates a random point in
the unit n-cube (well, with side lengths 2) and rejects any points
outside the unit n-sphere, then normalizes the length. This is correct
and works well at low dimension. However the volume of the unit
n-sphere compared to the unit n-cube drops exponentially. This method
eventually takes an extraordinary amount of time when dimensions get
past about 20, since virtually no samples are usable.

For example, here is the time taken to make pick 10 points as a
function of dimension up to 20:

1 : 11
2 : 1
3 : 0
4 : 1
5 : 0
6 : 1
7 : 1
8 : 17
9 : 4
10 : 3
11 : 13
12 : 32
13 : 15
14 : 41
15 : 220
16 : 897
17 : 1770
18 : 7426
19 : 48457
20 : 122647
...

It's equally correct, and much faster, to generate these points by
picking n standard Gaussians and normalizing. This method takes
negligible time even into thousands of dimensions.

Is this non-trivial enough that I might propose a patch? I wanted to
ask users first.

Thanks
Sean

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