Title: Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform
Speaker: Nir Ailon, Technion
Abstract:
Joint work with Edo Liberty
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 pojections. In particular, we generalize and use an intricate result by
Rudelson and Vershynin for sparse reconstruction which uses Dudley.s theorem
for bounding Gaussian processes. Our main result states that any set of
N = exp(O(n)) real vectors in n dimensional space can be linearly mapped to
a space of dimension k = O(logN 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 N = exp(O(n^{1/2})) achieved by Ailon and Liberty
and N = exp(O(N^{1/3})) by Ailon and Chazelle. The dependence in the distortion
constant however is believed to be suboptimal and subject to further
investigation. For constant distortion, this settles the open question posed by
these authors up to a polylog(n) factor while considerably simplifying their
construction.