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So, I'm not understanding something. If you take a vector formed only of ±1 coordinates, it's not going to be within the n-dimensional ball, nor on the n-1-dime

So, I'm not understanding something. If you take a vector formed only of ±1 coordinates, it's not going to be within the n-dimensional ball, nor on the n-1-dimensional sphere, it's going to be at the corner of the unit hypercube, right? Secondly, I don't understand by which measure the distribution of these vectors would be uniformly distributed or random. Normalizing the vectors to get the coordinates on the unit hypersphere is only ever going to land on 2^n different coordinates. Clearly in low dimension this wouldn't be a good way to select random points on the sphere… Does his approach make sense only because 2^300 is so much larger than 3 million?

Dylan16807 10y ago
> If you take a vector formed only of ±1 coordinates, it's not going to be within the n-dimensional ball, nor on the n-1-dimensional sphere The points are going to be on a hypersphere, just not the unit one. The radius will be about 17, right? > Does his approach make sense only because 2^300 is so much larger than 3 million? It has to be larger, I don't think it needs to be crushingly larger. As long as you're still sparse, all those extra digits on your random numbers don't matter. You can round it all the way to a single bit. Each dimension is only a tiny factor on your position on the sphere, so you get perfectly good coverage for something like dot products.
- jobigoud 10y ago
  Ah, thanks, got it now. Points are on the circumscribed sphere, radius sqrt(300).