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Suppose I want to generate points uniformly from a sphere (surface) of dimension $d$. A given solution is generate $d$ 1-dimensional gaussian points and then normalize the vector. Generating many points this way gives a distribution that is uniform over the surface of the sphere.

However " Note that once the vector is normalized, its coordinates are no longer statistically independent."

I didn't get the reasoning behind it. And a follow-up question is that how to generate iid points on a surface of sphere?

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The fact that $x_1, \ldots, x_n$ satisfy an equation $x_1^2 + \ldots + x_n^2$ means they can't be independent: if you know $x_2, \ldots, x_n$ then you know $x_1$ (up to sign).

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  • $\begingroup$ Yeah right. I thought they were telling that two vectors are not independent. Two different vectors generated this way are independent, right? So then why is there such a big deal about gaussians, like them being used everywhere? Why not do stuffs with spherical distribution? $\endgroup$ – the_dude Jul 20 '17 at 6:41
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The normalization forces the vector to be norm $1$, which makes the components depend on each other. The starkest example is the case $d=2$: if $X=(X_1,X_2)$ is standard bivariate Gaussian, and you consider $Z=X/\|X\|$, then $Z_1^2+Z_2^2=1$ almost surely, so $Z_1$ and $Z_2$ are obviously dependent.


The procedure that you described (normalizing a standard multivariate Gaussian) is one way to generate vectors from the uniform distribution on the unit sphere. I think this follows from the rotation invariance property of the Gaussian distribution.

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