Relaxed conditions for sparse signal recovery with general concave priors

The emerging theory of compressive or compressed sensing challenges the convention of modern digital signal processing by establishing that exact signal reconstruction is possible for many problems where the sampling rate falls well below the Nyquist limit. Following the landmark works of Candès et al. and Donoho on the performance of ℓ₁-minimization models for signal reconstruction, several authors demonstrated that certain nonconvex reconstruction models consistently outperform the convex ℓ₁-model in practice at very low sampling rates despite the fact that no global minimum can be theoretically guaranteed. Nevertheless, there has been little theoretical investigation into the performance of these nonconvex models. In this paper, a notion of weak signal recoverability is introduced and the performance of nonconvex reconstruction models employing general concave metric priors is investigated under this model. The sufficient conditions for establishing weak signal recoverability are shown to substantially relax as the prior functional is parameterized to more closely resemble the targeted ℓ₁ -model, offering new insight into the empirical performance of this general class of reconstruction methods. Examples of relaxation trends are shown for several different prior models.