Deep-plug-and-play proximal Gauss-Newton method with applications to nonlinear, ill-posed inverse problems

In this paper we propose a proximal Gauss-Newton method for the penalized nonlinear least squares optimization problem arising from regularization of ill-posed nonlinear inverse problems. By exploiting the modular structure that characterizes the proximal-type methods, we plug in a pre-trained graph neural net denoiser in place of the standard proximal map. This allows to mould the prior on the data. An encoder-decoder Graph U-Net architecture is proposed as denoiser, which works on unstructured data; its mathematical formulation is derived to analyse the Liptschitz condition. With the intent of showing the benefits of applying deep Plug-and-Play reconstructions, we consider as an exemplar application, the nonlinear Electrical Impedance Tomography, a promising non-invasive imaging technique mathematically formulated as a highly nonlinear ill-posed inverse problem.