Computer Science/Discrete Mathematics Seminar II

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 Lê Nguyen (Princeton).