I was studying this paper
Similarity Estimation Techniques from Rounding Algorithms by
Inside specifically had this statement.
Picking a random hyperplane amounts to choosing a normally distributed random variable for each dimension. Thus even representing a hash function in this family could require a large number of random bits. However, for n vectors, the hash functions can be chosen by picking O(log^2 n) random bits, i.e. we can restrict the random hyperplanes to be in a family of size 2^O(log^2 n)
I think it didnt make sense as what it means is that for each n vector, the dimensions would be
log^2 n. However, the equivalent statement is that the hyperplanes can be in a family of
size 2^O(log^2 n)
It doesnt seem to make any sense to me. Can someone explain to me why is this statement equals.