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Replying to and
Right but why are you drawing things from the vertices of a high d unit cube, that is one spiky and not symmetric object I refuse to believe in coordinate axes in concept space, thought is covariant
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Replying to and
If they’re sparse product elements would all be zero with high probability. It might even be true without a sparsity condition via central limit theorem. Ie likely as many product elements being positive as negative, and normally distributed in [-1, 1]
Replying to and
It's roughly Gaussian; mean 0, stddev=dim^{-1/2}. Some intuition: 1. Mean 0: symmetry. 2. Mean square = 1/d: if v1,...,vd form orthogonal basis then sum_i (vi,w)^2 =1 for any unit vector w. Now use symmetry (added d terms). 3. Gaussian: dot product is a sum of many small things
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