[jira] [Created] (MATH-931) Speed up UnitSphereRandomVectorGenerator for high dimensions

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[jira] [Created] (MATH-931) Speed up UnitSphereRandomVectorGenerator for high dimensions

AD_LB (Jira)
Sean Owen created MATH-931:
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             Summary: Speed up UnitSphereRandomVectorGenerator for high dimensions
                 Key: MATH-931
                 URL: https://issues.apache.org/jira/browse/MATH-931
             Project: Commons Math
          Issue Type: Improvement
    Affects Versions: 3.1.1
            Reporter: Sean Owen
            Priority: Minor
         Attachments: MATH-931.patch

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 in milliseconds 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.

Patch coming.

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