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The fact should say that the expected distance between two random points tends to the diameter as the dimension increases. The intuition is that to be close you
by dan-robertson 5y ago
The fact should say that the expected distance between two random points tends to the diameter as the dimension increases. The intuition is that to be close you need to be close in a large number of coordinates and the law of large numbers (though coordinates aren’t independent) suggests that is unlikely. If you fix one point on a sphere (say (1,0,…,0)) then, for a high dimension, most points will not have any extreme values in coordinates and will look like (~0,~0,…,~0) where ~0 means something close to zero. But if we sum the squares of everything apart from the first we get 1 - (~0)^2 ~= 1, so the distance from our fixed point is (1 - ~0)^2 + sum_2^n (0 - ~0)^2 ~= 1 + 1 = 2.
- Retric 5y agoYou forgot the square root on distance formula. Distance = square root of ((X1 - X2) ^ 2 + (Y1 - Y2) ^2 + …). Consider the origin (0,0,0, …) to a random point on the sphere (~0, ~0, ~0, …). So Distance from origin = square root of ((~0-0)^2 + (~0-0)^2 + (~0-0)^2 + … ), which sums to 1 by definition of the unit high dimensional sphere. Then plug in 1 vs 0 in the first place because we care about (1,0,0,0 …) and you get the correct answer = square root of ((~0-1)^2 + (~0-0)^2 + (~0-0)^2 + … ) ~= square root of 2. Edited to fix typo and add clarity.
- dan-robertson 5y agoWow. Can’t believe I missed that.