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by isoprophlex
612 days ago
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Pick an equator on an n-sphere. It is a hyperplane of dimensions (n-1) through the center, composed of all but one dimensions of your sphere. The xy plane for a unit sphere in xyz, for example. Uniformly distribute points on the sphere. For high n, all points will be very near the equator you chose. Obviously, in ofder for a point to be not close to this chosen equator, it projects close to 0 on all dimensions spanning the equatorial hyperplane, and not close to 0 on the dimension making up the pole-to-pole axis. |
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The analogy I have in mind is: if you throw n dice, for large n, the likelihood of one specific chosen dice being high value and the rest being low value is obviously rather small.
I guess that the consequence is still interesting, that most random points in a high-dimensional n-sphere will be close to the equator. But they will be close to all arbitrary chosen equators, so it's not that meaningful.
If the equator is defined as containing n-1 dimensions, then as n goes higher you'd expect it to "take up" more of the space of the sphere, hence most random points will be close to it. It is a surprising property of high-dimensional space, but I think it's mainly because we don't usually think about the general definition of an equator and how it scales to higher dimensions, once you understand that it's not very surprising.