  Incorporating prior information into inverse problems, e.g. via
maximum-a-posteriori estimation, is an important technique for facilitating
robust inverse problem solutions. In this paper, we devise two novel approaches
for linear inverse problems that permit problem-specific statistical prior
selections within the compound Gaussian (CG) class of distributions. The CG
class subsumes many commonly used priors in signal and image reconstruction
methods including those of sparsity-based approaches. The first method
developed is an iterative algorithm, called generalized compound Gaussian least
squares (G-CG-LS), that minimizes a regularized least squares objective
function where the regularization enforces a CG prior. G-CG-LS is then
unrolled, or unfolded, to furnish our second method, which is a novel deep
regularized (DR) neural network, called DR-CG-Net, that learns the prior
information. A detailed computational theory on convergence properties of
G-CG-LS and thorough numerical experiments for DR-CG-Net are provided. Due to
the comprehensive nature of the CG prior, these experiments show that DR-CG-Net
outperforms competitive prior art methods in tomographic imaging and
compressive sensing, especially in challenging low-training scenarios.
