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Comment by okintheory | Hacker News Reader

Yes, this can be made precise: For example, consider a uniform distribution on a d-dimensional cube with side length 1. Only a (1/2)^d fraction of the mass is within 0.25 distance of the average in every coordinate.

In high dimensions, almost all the mass is "near the boundary" in at least one coordinate.

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Or as one of my professor would say (after showing the calculation for n-dimensional spheres): "and that's why infinite dimensional oranges are all skin!"

This must be a common saying. I've heard it from every lecturer I've ever had for machine learning and sampling techniques.

Lecturers get together in a staff room early in the morning and rehearse the delivery corny jokes in unison, the way teams of shop clerks in Japan go over phrases like "irasshaimase" and "omatase shimashita".

It's undoubtedly pithy. (sorry!)

I suspect it comes from statistical physics, as it's a very common approximation when you're in the thermodynamical limit, which basically means that the phase space has a very large number of dimensions.

Weirdly enough, even the proportion of points within a distance of 1/2 to the midpoint becomes really small for higher dimensions.

takeaway: With enough dimensions everyone is weird

Bad news for advertisers who want to use a zillion targeting factors -- and yet somehow not be disappointed by the minuscule audience that results ...

Or -- if you think everybody is the same -- then you aren't looking hard enough!

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