  Sparse recovery principles play an important role in solving many nonlinear
ill-posed inverse problems. We investigate a variational framework with support
Oracle for compressed sensing sparse reconstructions, where the available
measurements are nonlinear and possibly corrupted by noise. A graph neural
network, named Oracle-Net, is proposed to predict the support from the
nonlinear measurements and is integrated into a regularized recovery model to
enforce sparsity. The derived nonsmooth optimization problem is then
efficiently solved through a constrained proximal gradient method. Error bounds
on the approximate solution of the proposed Oracle-based optimization are
provided in the context of the ill-posed Electrical Impedance Tomography
problem. Numerical solutions of the EIT nonlinear inverse reconstruction
problem confirm the potential of the proposed method which improves the
reconstruction quality from undersampled measurements, under sparsity
assumptions.
