3 ms·
I was going to comment that what's going on here doesn't have much to do with the Gaussian distribution. In high dimensions, almost all of the volume of the uni
by woopwoop 9y ago
I was going to comment that what's going on here doesn't have much to do with the Gaussian distribution. In high dimensions, almost all of the volume of the unit ball is concentrated near the unit sphere. In the first comment, Frank Morgan makes the same remark, pointing out that you get the same effect with the uniform distribution on the unit cube in high dimensions.
High dimensions are weird.
- paulgb 9y agoIt's true! I just made a colab notebook to visualize the effect: https://colab.research.google.com/notebook#fileId=1znFHwemxaNf2x8iAHWonBa_2LsKSGaZD https://colab.research.google.com/notebook#fileId=1znFHwemxa... (seems to require a Google account, sorry in advance)
- semi-extrinsic 9y agoInstead of plotting the cumulative mass distribution, why not just plot the mass distribution itself? I.e. the derivatives of the curves you've plotted?
- tomas_fiers 9y agoGreat, thanks!
- Sharlin 9y agoIn retrospect it's pretty trivial to see why it is so: it directly results from the basic fact that the hypervolume of an n-body grows as the nth power of linear size.
- dwaltrip 9y agoHow much of this stuff has fundamental implications for the hardest problems that scientists, researchers, and policy makers are facing today? My intuition says many of those problems have high dimensionality, but I'm not really confident about my intuition here.
- trhway 9y agothis is probably why such things like "average" person (or any other object with multiple characteristics) practically don't really exist. Everybody is [close to] exceptional in at least some of their characteristics or their combination :)