THE FAST JOHNSON-LINDENSTRAUSS TRANSFORM AND APPROXIMATE NEAREST NEIGHBORS

We introduce a new low-distortion embedding of l(2)(d) into 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.