Learning to Regularize Using Neumann Networks

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, inpainting, compressed sensing, and superresolution all fit in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term and a regularizer which promotes desirable properties in the solution. Recent advances have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method, which directly solves the linear inverse problem with a data-driven nonlinear regularizer via a truncated Neumann series. This Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. In addition, when the images belong to a union of subspaces, we prove under appropriate assumptions there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem.