5 ms·
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
by okintheory 11y ago
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.
- carlob 11y agoOr 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!"
- kirrent 11y agoThis must be a common saying. I've heard it from every lecturer I've ever had for machine learning and sampling techniques.
- carlob 11y agoI 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.
- mattkrause 11y agoIt's undoubtedly pithy. (sorry!)
- kazinator 11y agoLecturers 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".
- contravariant 11y agoWeirdly enough, even the proportion of points within a distance of 1/2 to the midpoint becomes really small for higher dimensions.
- foota 11y agotakeaway: With enough dimensions everyone is weird