Analysis of the ratio of ℓ₁ and ℓ₂ norms in compressed sensing

We study the ratio of ℓ₁ and ℓ₂ norms (ℓ₁/ℓ₂) as a sparsity-promoting objective in compressed sensing. We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the ℓ₁/ℓ₂ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare ℓ₁/ℓ₂ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach support selection, and we demonstrate that it empirically improves the performance of existing ℓ₁/ℓ₂ algorithms.