  The recovery of sparse generative models from few noisy measurements is an
important and challenging problem. Many deterministic algorithms rely on some
form of $\ell_1$-$\ell_2$ minimization to combine the computational convenience
of the $\ell_2$ penalty and the sparsity promotion of the $\ell_1$. It was
recently shown within the Bayesian framework that sparsity promotion and
computational efficiency can be attained with hierarchical models with
conditionally Gaussian priors and gamma hyperpriors. The related Gibbs energy
function is a convex functional and its minimizer, which is the MAP estimate of
the posterior, can be computed efficiently with the globally convergent
Iterated Alternating Sequential (IAS) algorithm \cite{CSS}. Generalization of
the hyperpriors for these sparsity promoting hierarchical models to generalized
gamma family yield either globally convex Gibbs energy functionals, or can
exhibit local convexity for some choices for the hyperparameters. \cite{CPrSS}.
The main problem in computing the MAP solution for greedy hyperpriors that
strongly promote sparsity is the presence of local minima. To overcome the
premature stopping at a spurious local minimizer, we propose two hybrid
algorithms that first exploit the global convergence associated with gamma
hyperpriors to arrive in a neighborhood of the unique minimizer, then adopt a
generalized gamma hyperprior that promote sparsity more strongly. The
performance of the two algorithms is illustrated with computed examples.
