Intuition question. You draw two vectors from an n-dimensional isotropic Gaussian where n>>1. What is the typical angle between them?
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If the Gaussian is centered at zero, won’t both angles be uniformly distributed on the circle, and thus have undefined expected value? And doesn’t this imply the expectation of the difference is undefined?
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yes. but n>>1 ;)
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If I understand your question correctly this simplifies to computing the dot product of two gaussian vectors, which we need to define the cosine similarity. The dot product is a sum of i.i.d. random variables so the expectation doesn’t depend on the dimension n.
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