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I'm using a Gaussianity assumption over 500-dimensional data in my work and I wanted to check the validity of my assumption. I developed a transformation that relies on this assumption and I have good results (I work on face recognition) that's why I wanted some statistical validation or if it can be improved using for example heavy-tailed priors. I've been looking for Gaussianity tests in Google scholar and I've found far too many techniques.

So my question is, which techniques are considered "standard" or most used for these kinds of problems?

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    $\begingroup$ In comparison with the 1 dimensional case, are you only looking for a hypothesis test like the Shapiro Wilk test or are you looking for something like a QQ plot to demonstrate approximate Gaussianity? $\endgroup$
    – Hugh
    Oct 3, 2016 at 10:17
  • $\begingroup$ Indeed, I'm looking for something like the Shapiro Wilk test. $\endgroup$
    – vphenix
    Oct 5, 2016 at 11:53
  • $\begingroup$ The comment about normality testing being useless is true, but, if you absolutely must do a hypothesis test (for some reason), the MVN package in R has some hypothesis tests for this. $\endgroup$
    – Dave
    Jul 20, 2021 at 16:20

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I assume you want to say multi-gaussian assumption. It is quite improbable that will be true in so high dimension, so here is a way to approach your problem. Try independent component analysis, see Making sense of independent component analysis. This will in practice search for linear combinations that are as far away from a normal distribution as possible.

If you want this could maybe be turned into an hypothesis test, but before doing that look at Is normality testing 'essentially useless'?

Tests for multivariate normality—a critical review with emphasis on weighted 𝐿2-statistics is a relatively new paper reviewing various tests, with an accompanying R package mnt. It also gives references to earlier works, and works looking at visualization methods.

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