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There are two reasons to focus on "gaussianity". (1) Orthogonal transformations of gaussian distributions are again gaussian. (2) Mixing of signals tends to a gaussian distribution via Central Limit Theorem. The latter is very good reason, but I would like to focus on the correctness of the motivation in (1).

The Gaussian distribution is hardly the only distribution preserved under orthogonal transformations. Take any distribution, and consider its average over possible rotations. The uniform distribution on the set |x| < R fashions a non-exotic concrete example. All of these distributions should be untenable for independent component analysis (ICA).

Given that the Gaussian distribution is but one of many possible theoretical "problem distributions" that ICA should have difficulty with, (A)why all the fixation on the Gaussian distribution, and (B) does this pose practical concerns?

I was surprised not to find any tutorial that mentions this.

Relating to (B), It's possible that most measures of "nongaussianity" e.g. kurtosis are also measures of "non-rotation-invariance", but I would like a second opinion on this.

hardmath
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user3391229
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  • You can average the distribution via the described procedure, but will it be the distribution from the same class ? – Evgeny Sep 28 '15 at 17:08
  • Sorry, I'm sure that was confusing. If you want to get a rotationally symmetric distribution and you're not sure were to start, you can start with a non-rotationally symmetric distribution, rotate it a bunch, and average what you get. I just mean to emphasize that rotationally-symmetric distributions are many and are not limited to gaussian distributions. – user3391229 Sep 28 '15 at 17:10
  • One moment, let me edit. I see how this is confusing. – user3391229 Sep 28 '15 at 17:16
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    According to Maxwell's theorem, Gaussian distributions on $\mathbb R^n$ are the only ones that are spherically symmetric and have independent components. One should add that among Gaussian distributions on $\mathbb R^n$ there are some whose components are neither independent nor identically distributed. ${}\qquad{}$ – Michael Hardy Sep 28 '15 at 17:38

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