Sparsity Promoting Hybrid Solvers for Hierarchical Bayesian Inverse Problems

Abstract

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 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 convexity for some choices for the hyperparameters [D. Calvetti et al., Inverse Problems, 36 (2020), 025010]. 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 and then adopt a generalized gamma hyperprior that promotes sparsity more strongly. The performance of the two algorithms is illustrated with computed examples.