gaussian random vectors are approximately uniform on the hollow sphere

[[concept]]

Topics

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Take Away

$\mathcal{N}(0,I_d)\approx \text{Unif}({\mathbb{S}}^{d-1}(\sqrt{d}))$

informal approximation!

This is an

We get this by noting

1. When $x\sim \mathcal{N}(0,I_d)$, we have $\text{Law}\left(\frac{x}{||x||}\right)=\text{Unif}({\mathbb{S}}^{d-1})$ ( which we get from normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere )
2. $||x||\approx \sqrt{d}$, which follows from the concentration inequality for magnitude of standard gaussian random vector Combining the two yields the heuristic approximation.

This idea is formalized in Borel’s central limit theorem

References

References

See Also

Mentions

Mentions

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