Almost Euclidean subspaces of ℓ1N VIA expander codes

AbstractWe give an explicit (in particular, deterministic polynomial time) construction of subspaces X⊆ℝN of dimension (1−o(1))N such that for every x∈X, $$ (\log N)^{ - O(\log \log \log N)} \sqrt N \left\| x \right\|_2 \leqslant \left\| x \right\|_1 \leqslant \sqrt N \left\| x \right\|_2 $$. If we are allowed to use N1/log logN ⩽ No(1) random bits and dim(X) ⩾ (1−η)N for any fixed constant η, the lower bound can be further improved to $$ (\log N)^{ - O(1)} \sqrt N \left\| x \right\|_2 $$.Through known connections between such Euclidean sections of ℓ1 and compressed sensing matrices, our result also gives explicit compressed sensing matrices for low compression factors for which basis pursuit is guaranteed to recover sparse signals. 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.