  Many challenging image processing tasks can be described by an ill-posed
linear inverse problem: deblurring, deconvolution, inpainting, compressed
sensing, and superresolution all lie in this framework. Traditional inverse
problem solvers minimize a cost function consisting of a data-fit term, which
measures how well an image matches the observations, and a regularizer, which
reflects prior knowledge and promotes images with desirable properties like
smoothness. Recent advances in machine learning and image processing 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 of solving inverse problems inspired by the Neumann series,
which we call a Neumann network. Rather than unroll an iterative optimization
algorithm, we truncate a Neumann series which directly solves the linear
inverse problem with a data-driven nonlinear regularizer. The 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. Finally, when the images belong to a union of
subspaces and under appropriate assumptions on the forward model, we prove
there exists a Neumann network configuration that well-approximates the optimal
oracle estimator for the inverse problem and demonstrate empirically that the
trained Neumann network has the form predicted by theory.
