Sparsity Lower Bounds for Dimensionality Reducing Maps

Computer Science/Discrete Mathematics Seminar II
Topic:	Sparsity Lower Bounds for Dimensionality Reducing Maps
Speaker:	Jelani Nelson
Affiliation:	Member, School of Mathematics
Date:	Tuesday, January 22
Time/Room:	10:30am - 12:30pm/S-101

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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).