Sparser Johnson-Lindenstrauss Transforms
Abstract
We give two different and simple constructions for dimensionality reduction in via linear mappings that are sparse: only an -fraction of entries in each column of our embedding matrices are non-zero to achieve distortion with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar, and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up applications where dimensionality reduction is used.
Keywords
Cite
@article{arxiv.1012.1577,
title = {Sparser Johnson-Lindenstrauss Transforms},
author = {Daniel M. Kane and Jelani Nelson},
journal= {arXiv preprint arXiv:1012.1577},
year = {2014}
}
Comments
v6: journal version, minor changes, added Remark 23; v5: modified abstract, fixed typos, added open problem section; v4: simplified section 4 by giving 1 analysis that covers both constructions; v3: proof of Theorem 25 in v2 was written incorrectly, now fixed; v2: Added another construction achieving same upper bound, and added proof of near-tight lower bound for DKS scheme