discussion following https://github.com/VectorDB-NTU/Extended-RaBitQ/issues/3

[chimborazo11]

'a random vector follows uniform distribution on the unit sphere', I wonder is based on the before-rotation vectors are uniformly distributed(since the random rotation matrix is shared by all vectors!). Otherwise, assume all vectors are concentrated in 1 dimension and stay very close, after applying a random rotation matrix on them, they still possibly stay in 1 dimension and the theorem can not hold.

[gaoj0017]

Sorry for the delayed reply.

Here we have two different concepts: (1) the distirbution of every vector; and (2) the correlation between different vectors.

The claim is for (1): every vector follows uniform distribution on the unit sphere. On the other hand, they can be correlated to each other (for example, all vectors are concentrated in 1 dimension and stay very close, as you mentioned). They are compatible with each other.