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.

翻译：稀疏恢复原理在解决许多非线性病态逆问题中发挥着重要作用。我们提出了一种基于支撑集先验的变分框架，用于处理压缩感知中的稀疏重建，其中可获得的测量是非线性的且可能受到噪声污染。我们设计了一种名为Oracle-Net的图神经网络，用于从非线性测量中预测支撑集，并将其整合到正则化恢复模型中，以强制执行稀疏性。随后，通过约束近端梯度法高效求解由此产生的非光滑优化问题。在病态电阻层析成像问题的背景下，我们给出了所提基于Oracle先验优化近似解的误差界。电阻层析成像非线性逆重建问题的数值解验证了所提方法的潜力，该方法在稀疏性假设下提高了欠采样测量的重建质量。