NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/comp/theory/2357

Internet Message Format | 1992-11-08 | 2.6 KB

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From: [e-mail] (T. Scott Thompson)
Subject: Re: Uniform noise in a d-sphere
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Organization: Economics Department, University of Minnesota
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Date: Fri, 6 Nov 1992 18:03:53 GMT
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[e-mail] (Michael Murphy) writes:


>I am trying to compute uniformly random noise inside a d-dimensional
>sphere.  I have identified two ways of doing so:

[stuff about how to generate uniform noise deleted.]

>I believe that method two works.  However, when I project a large number of
>points in a high dimensional space (e.g. 25) onto the plane (by ignoring all
>but two coordinates), I get something that resembles a square.  My advisor
>is not so sure that this is correct and I can only give an intuitive
>argument as to why it may be true.

>Consider a set of 10,000 points inside a ball (3-d).  Projecting this ball
>into the x,y plane, one would expect to see something whose convex hull
>resembles a circle, but with more points in the middle than on the sides
>because the ball is fatter in the middle.

>I claim that as the dimension increases, one would expect this middle section
>to get more and more dense, with the bulk of the points falling inside the
>largest square inscribable in the circle.  I would like to try the Monte Carlo
>method as a sanity check, but it take too long to generate points in a high
>dimensional space.

Other posters have commented on the validity of the generation
methods, but I haven't yet seen any response to this argument about
how to check the output.

I think that your claim is invalid.  If your distribution on the unit
ball is radially symmetric, as the uniform distribution must be, then
so should be the "projection" that you describe.  That is, the
distribution of (x_1,x_2) should be unaffected by rotations of R^d.
But some rotations of R^d correspond exactly to rotating (x_1,x_2) in
the plane, leaving other coordinates invariant.  So your projection to
R^2 should be invariant to rotations of the plane.  Clearly a square
does not satisfy this criteria, so your claim cannot be correct.
--
T. Scott Thompson              email:  [e-mail]
Department of Economics        phone:  [phone]
University of Minnesota        fax:    [phone]