locality sensitive hashing is the typical method here. You generate a ton of random vectors. These vectors automatically give rise to infinite hyperplanes (the one they are normal to). Each vector/hyperplane is associated with a bit in the final hash. Then, each input vector is hashed by setting the bit if it's on one side of the hyperplane or unsetting it if it's on the other. The hamming distance between two vectors is now correlated with the cosine similarity. Or something like that.