The fast johnson-lindenstrauss transform and approximate nearest neighbors

We introduce a new low-distortion embedding of l ₂ ^d l _p ^O(log n (p = 1, 2) called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the "Heisenberg principle" of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in l ₁ and l ₂. We consider the case of approximate nearest neighbors in l ₂ ^d. We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.