yesterday i tried to understand the thing people mean when they say that high-dimensional balls and cubes are "spiky." it seems to me that we can be much more precise than the usual calculations people do here, with very little additional effort. might write a blog post
Conversation
Replying to
in the meantime this blog post is cute and the comment about a multivariate gaussian in high dimensions looking like a donut was super helpful
in high dimensions a multivariate gaussian looks surprisingly similar to a uniform distribution on a sphere
observablehq.com/@tophtucker/th
2
12
the surface of the n-cube [-1, 1]^n is the set of points where at least one coordinate is 1 or -1. the % of points all of whose coordinates are at least x/2n away from 1 or -1 is (1 - x/n)^n ~ e^{-x}. so almost all points have at least one coordinate within O(1/n) of 1 or -1
1
2
this just reflects the idea that a random point in the n-cube is a sequence of n iid samples from the uniform distribution on [-1, 1], so you expect the coordinates to be uniformly distributed and in particular for O(1) of them to be within O(1/n) of the ends
6
Replying to
I recently discussed this point about gaussians in a talk to my patrons, and found that there wasn't a blog post that made this point (and the one about balls & cubes) in the way that I'd have wanted to, so I hope you write it!
2
Replying to
weird, just did a homework set involving these
confused me for a bit that, while the volume goes into the corners of a cube as d increases, for any *fixed* epsilon corner the volume goes to 0. hypercubes get spiky but the volume is in the *spikes*, not the tips of the corners
1
i'm kind of a math brainlet though, so i think this is less interesting/weird than i thought when i was doing it out