Sparse signal recovery with additional ℓ2 null space constraint

This paper studies a relaxed version of the analysis sparsity model, in which the signal produces an output vector that is not rigorously sparse itself, instead it is within a ℓ₂ distance from a sparse vector. Conversely, this can also be viewed as a synthesis model with the additional requirement that the sparse decomposition has only a limited component in the dictionary´s null space. We show that if this ℓ₂ constraint is sufficiently tight, a sparse signal can be recovered via ℓ₀ minimization if the Restricted Isometry constant of the system matrix satisfies δ_2k-1 <; 1, which is an improvement over the δ_2k <; 1 condition used in the usual synthesis sparse model. In practical simulations, the mixture of sparsity and ℓ₂ constraints leads to reduced recovery errors when sparsity alone is not enough.