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The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. To do so, it requires random variables to be sampled from a normal distribution. In this proof, it states that the normal distribution is used because a normalized vector of Gaussian variables is uniformly distributed on the surface of a d-dimensional sphere. The proof of this can be found here.

Do any other distributions have this same property such that they can be used in the dimensionality reduction?

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