Almost Euclidean subspaces of \ell_1^N via expander codes
Metric Geometry
2009-03-26 v2 Functional Analysis
Abstract
We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log N}. If we are allowed to use N^{o(1)} random bits, this factor can be improved to poly(log N). Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.
Keywords
Cite
@article{arxiv.0709.0887,
title = {Almost Euclidean subspaces of \ell_1^N via expander codes},
author = {Venkatesan Guruswami and James R. Lee and Alexander Razborov},
journal= {arXiv preprint arXiv:0709.0887},
year = {2009}
}