Almost Euclidean subspaces of \ell_1^N via expander codes

TL;DR

This paper presents a deterministic polynomial-time method to construct high-dimensional subspaces of R^N where the  and scaled  norms are nearly equivalent, using expander graphs inspired by error-correcting codes.

Contribution

The authors provide the first explicit construction of almost Euclidean subspaces of ^N with near-optimal dimension, improving previous probabilistic methods.

Findings

Constructed subspaces of dimension (1-o(1))N with norm equivalence up to (log N)^{log log log N}

Achieved near Euclidean structure using explicit, deterministic methods

Improved factor to poly(log N) with limited randomness (N^{o(1)} bits)

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.