Almost-Euclidean subspaces of ℓ1N via tensor products: A simple approach to randomness reduction

Abstract

It has been known since 1970's that the N-dimensional ℓ1- space contains almost Euclidean subspaces whose dimension is Ω(N). However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any γ>0, allows us to exhibit almost Euclidean Ω(N)-dimensional subspaces of ℓ1N while using only Nγ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding almost Euclidean subspaces with arbitrarily small distortions. © 2010 Springer-Verlag.