Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform

We introduce a new low-distortion embedding of l2d into lpO(log n) (p=1,2), called the Fast-Johnson-Linden-strauss-Transform. 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, ie, its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in l1 and l2. We consider the case of approximate nearest neighbors in l2d. We provide a faster algorithm using classical projections, which we then further speed up by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.