Efficient sparsity-promoting MAP estimation for Bayesian linear inverse problems

Bayesian hierarchical models can provide efficient algorithms for finding sparse solutions to ill-posed linear inverse problems. The models typically comprise a conditionally Gaussian prior model for the unknown augmented by a generalized gamma hyper-prior model for the variance hyper-parameters. This investigation generalizes such models and their efficient maximum a posterior estimation using the iterative alternating sequential algorithm in two ways: (1) general sparsifying transforms: diverging from conventional methods, our approach permits use of sparsifying transformations with nontrivial kernels; (2) unknown noise variances: the noise variance is treated as a random variable to be estimated during the inference procedure. This is important in applications where the noise estimate cannot be accurately estimated a priori. Remarkably, these augmentations neither significantly burden the computational expense of the algorithm nor compromise its efficacy. We include convexity and convergence analysis and demonstrate our method's efficacy in several numerical experiments.