Almost-Euclidean subspaces of ${\mathrm{\ell }}_1^N$ via tensor products: a simple approach to randomness reduction

It has been known since 1970’s that the N-dimensional ${\mathrm{\ell }}_1$-space contains nearly Euclidean subspaces whose dimension is $\mathrm{\Omega }(N)$. 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 $a>0$, allows to exhibit nearly Euclidean $\mathrm{\Omega }(N)$-dimensional subspaces of ${\mathrm{\ell }}_1^N$ while using only $N^a$ 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.