Sparsity promoting hybrid solvers for hierarchical bayesian inverse problems

The recovery of sparse generative models from few noisy measurements is an important and challenging problem. Many deterministic algorithms rely on some form of ℓ1-ℓ2 minimization to combine the computational convenience of the ℓ2 penalty and the sparsity promotion of the ℓ1. 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 maximum a posteriori (MAP) estimate of the posterior, can be computed efficiently with the globally convergent Iterated Alternating Sequential (IAS) algorithm [D. Calvetti, E. Somersalo, and A. Strang, Inverse Problems, 35 (2019), 035003]. Generalization of the hyperpriors for these sparsity promoting hierarchical models to a generalized gamma family either yield globally convex Gibbs energy functionals or can exhibit local c...