I think I get part of the N-ball analogy, but I'm trying to piece together the rest...
So it seems like one problem/feature with high dimensional spaces is that you have more regions which are orthogonal to one other. So, it's possible to have quite a bit of motion, which to a human mind might give the impression that one is doing quite a lot—but then you might discover a new way of turning around and find that space extends in another direction, too, at which point you realize you covered a smaller fraction of the available space than you previously believed.
What I don't get is how the volume of an N-ball with fixed radius going to zero as N goes to infinity plays into the analogy. When you say, "when you compare it to a hypercube of the same dimensional count," I don't get what in the programming analogy would link to the N-ball rather than the high dimensional hypercube. I also get the algebraic justification for this property of N-balls, but I still have no real geometric intuition for why it would be the case.
Could you clarify what's meant by, "as the number of dimensions increases the closer the 'halfway' point of the volume of the N-ball gets to the edge of the N-ball"? Not sure what's meant by 'halfway' point here, or edge of the N-ball. My guess is that you're talking about distribution of volume, like if you have a tapered cup that gets small at the bottom, most of the volume will be at the top.
So it seems like one problem/feature with high dimensional spaces is that you have more regions which are orthogonal to one other. So, it's possible to have quite a bit of motion, which to a human mind might give the impression that one is doing quite a lot—but then you might discover a new way of turning around and find that space extends in another direction, too, at which point you realize you covered a smaller fraction of the available space than you previously believed.
What I don't get is how the volume of an N-ball with fixed radius going to zero as N goes to infinity plays into the analogy. When you say, "when you compare it to a hypercube of the same dimensional count," I don't get what in the programming analogy would link to the N-ball rather than the high dimensional hypercube. I also get the algebraic justification for this property of N-balls, but I still have no real geometric intuition for why it would be the case.
Could you clarify what's meant by, "as the number of dimensions increases the closer the 'halfway' point of the volume of the N-ball gets to the edge of the N-ball"? Not sure what's meant by 'halfway' point here, or edge of the N-ball. My guess is that you're talking about distribution of volume, like if you have a tapered cup that gets small at the bottom, most of the volume will be at the top.