An Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform

The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Veshynin [2008] for sparse reconstruction which uses Dudley's theorem for bounding Gaussian processes. Our main result states that any set of N = exp((O) over tilde (n)) real vectors in n dimensional space can be linearly mapped to a space of dimension k = O(log N polylog(n)), while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time O(n log n) on each vector. This improves on the best known bound N = exp((O) over tilde (n(1/2))) achieved by Ailon and Liberty [2009] and N = exp((O) over tilde (n(1/3))) by Ailon and Chazelle [2010]. The dependence in the distortion constant however is suboptimal, and since the publication of an early version of the work, the gap between upper and lower bounds has been considerably tightened obtained by Krahmer and Ward [2011]. For constant distortion, this settles the open question posed by these authors up to a polylog(n) factor while considerably simplifying their constructions.