| Neat thank you for creating this! That said I think someone who cares about TurboQuant probably already has a bit of linear algebra knowledge. While the initial review section is definitely appreciated, I don't think it's going to help much since you need a decent level of "mathematical maturity" anyway. The "Coordinates of a random unit vector are all small" had me scratching my head a bit, and the language is a bit misleading since it's actually that the expected variance of any individual component is 1/N (it can't be that every coordinate is close to ±1/sqrt{N} because the mean of any individual component is clearly 0 by symmetry). So that one should probably use more explanation since I had to work through it myself: Denoting the random unit vector {X1 ... Xn}, this is a point on a hypersphere: * Sum[x_i ^2] = 1 (unit vector condition) * E[Sum[X_i ^2]] = 1 (expectation of both sides) * Sum[E[X_i ^2]] = 1 (linearity of expectation) * E[X_1 ^2] = 1/N (by rotational symmetry E[X_1 ^2] = E[X_2 ^2] = ..) I don't think you can make the stronger claim that E[|X_1|] = 1/sqrt{N} since that's using L1 norm on a single component, so it'd be more correct to say the RMS is just the standard deviation of the components. And this fits with the intuition that in high dimensional space has "spiky" hypercubes with the hypersphere inscribed in it close to the origin. |