I want to assess distances between pairs of high-dimensional vectors (~1500 features). My vectors have been normalized by their L2 norms, so they all have unit length and point to the surface of a hypershpere.
I know of a particular instance of the "curse of dimensionality", which says that with increasing dimensions the difference between minimum and maximum distances tends to zero, and Euclidean distance becomes a useless metric. However, does this problem also hold in the particular case of normalized vector lengths? I've seen discussions noting that the bulk of the volume in high-dimensional spaces in concentrated in the outer shell, which is precisely the situation I'm dealing with. Does Euclidean distance somehow become "valid" again?
Alternatively, I would say that the distance between two vectors touching the surface of a hypersphere is simply theta/2*pi*r, where theta is the angle between vectors in radians, and the radius r is 1. (Consider a 2D circle where the only way from point A to B is by traversing the surface of the circle.) However, here too I am uncertain whether this simple intuition carries over to higher dimensions or whether unexpected behavior occurs.