  In this paper, we propose a novel sparse recovery method based on the
generalized error function. The penalty function introduced involves both the
shape and the scale parameters, making it very flexible. The theoretical
analysis results in terms of the null space property, the spherical section
property and the restricted invertibility factor are established for both
constrained and unconstrained models. The practical algorithms via both the
iteratively reweighted $\ell_1$ and the difference of convex functions
algorithms are presented. Numerical experiments are conducted to illustrate the
improvement provided by the proposed approach in various scenarios. Its
practical application in magnetic resonance imaging (MRI) reconstruction is
studied as well.
