Discussion Topic: (mis)intuitions about High dimensional space..

["Subbarao Kambhampati" <rao@asu.edu>]

Here are two interesting facts. 

1. High dimensional apples are all peel and no core. Specifically, consider spherical apples of
n dimensions. Suppose you were to peel off an epsilon-think apple skin off the apple, what is the
fraction of the apple that is left? If you consider unit radius apples (spheres) you can show that 
the percentage of apple left is 97% in 3-D, and 0.004% in 1000-D (i.e., thousand dimensional space). (You can
assume that the volume of an n-dimensional sphere is O( r^n) where r is the radius.


2. In high-dimensions, any randomly chosen pair of vectors are, with very high probability, perpendicular
to each other. 

You are welcome--on the blog--to (a) try to prove them
and/or (b) discuss why any of these are relevant to information retrieval using vector space
ranking.

Rao

ps: If intellectual adventure/stimulation is not enough of an incentive to post on the blog, 
    note that blog participation is viewed as class participation--and part of your grade for this course will be participation based.