  Since most inverse problems arising in scientific and engineering
applications are ill-posed, prior information about the solution space is
incorporated, typically through regularization, to establish a well-posed
problem with a unique solution. Often, this prior information is an assumed
statistical distribution of the desired inverse problem solution. Recently, due
to the unprecedented success of generative adversarial networks (GANs), the
generative network from a GAN has been implemented as the prior information in
imaging inverse problems. In this paper, we devise a novel iterative algorithm
to solve inverse problems in imaging where a dual-structured prior is imposed
by combining a GAN prior with the compound Gaussian (CG) class of
distributions. A rigorous computational theory for the convergence of the
proposed iterative algorithm, which is based upon the alternating direction
method of multipliers, is established. Furthermore, elaborate empirical results
for the proposed iterative algorithm are presented. By jointly exploiting the
powerful CG and GAN classes of image priors, we find, in compressive sensing
and tomographic imaging problems, our proposed algorithm outperforms and
provides improved generalizability over competitive prior art approaches while
avoiding performance saturation issues in previous GAN prior-based methods.
