Counterintuitive Properties of High Dimensional Space (2018)

The distance between two uniform random points on an n-sphere clusters around the equator. The article shows a histogram of the distribution in fig. 11. While it looks Gaussian, it is more closely related to the Beta distribution. I derived it in my notes, as (surprisingly) I could not find it easily in literature:

https://xn--2-umb.com/21/n-sphere

beautiful visualizations, how did you make them?

> The distance between two uniform random points on an n-sphere clusters around the equator.

This sentence makes no sense to me.

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.

He means it clusters around the distance from a pole to the equator.

Correct. I was too short in my comment. It's explained in the article: without loss of generality you can call one of the two points the 'north pole' and then the other one will be distributed close to the equator.

The first one IIRC with Geogebra, all the rest with Matplotlib. The design goal was to maximize on 'data-ink ratio'.

My first thought is that it's rather obvious, but I'm probably wrong, can you help me understand?

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.

"clusters" is acting as a verb here, not a noun.

> 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.

You're exactly right, this whole thing is indeed a bit of an obvious nothingburger.