Jelani Nelson

Member, School of Mathematics

January 22, 2013

Abstract:

We give near-tight lower bounds for the sparsity required in several dimensionality reducing linear maps. In particular, we show:

(1) The sparsity achieved by [Kane-Nelson, SODA 2012] in the sparse Johnson-Lindenstrauss lemma is optimal up to a log(1/eps) factor.

(2) RIP_2 matrices preserving k-space vectors in R^n with the optimal number of rows must be dense as long as k < n / polylog(n).

(3) Any oblivious subspace embedding with 1 non-zero entry per column and preserving d-dimensional subspaces in R^n must have Omega(d^2) rows, matching an upper bound of [Nelson-Nguyen, 2012] for constant distortion.

Joint work with Huy LĂȘ Nguyen (Princeton).