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 ℓ[subscript 1]-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 ℓ[subscript 1][superscript N] while using only N γ 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.