Almost-Euclidean subspaces of $\ell_1^N$ via tensor products: a simple approach to randomness reduction

TL;DR

This paper introduces a simple tensor product-based method to construct nearly Euclidean subspaces of high-dimensional $\, ext{l}_1^N$ using significantly fewer random bits, with applications in computational problems.

Contribution

It provides a low-tech, probabilistic construction of large nearly Euclidean subspaces in $\, ext{l}_1^N$ using only polynomially many random bits, extending prior work.

Findings

Constructs nearly Euclidean subspaces with $\, ext{Omega}(N)$ dimension

Uses only $N^a$ random bits for any $a > 0$

Achieves arbitrarily small distortions in subspace embeddings

Abstract

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. 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 $\Omega(N)$-dimensional subspaces of $\ell_1^N$ 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.