  Solving inverse problems is a fundamental component of science, engineering
and mathematics. With the advent of deep learning, deep neural networks have
significant potential to outperform existing state-of-the-art, model-based
methods for solving inverse problems. However, it is known that current
data-driven approaches face several key issues, notably hallucinations,
instabilities and unpredictable generalization, with potential impact in
critical tasks such as medical imaging. This raises the key question of whether
or not one can construct deep neural networks for inverse problems with
explicit stability and accuracy guarantees. In this work, we present a novel
construction of accurate, stable and efficient neural networks for inverse
problems with general analysis-sparse models, termed NESTANets. To construct
the network, we first unroll NESTA, an accelerated first-order method for
convex optimization. The slow convergence of this method leads to deep networks
with low efficiency. Therefore, to obtain shallow, and consequently more
efficient, networks we combine NESTA with a novel restart scheme. We then use
compressed sensing techniques to demonstrate accuracy and stability. We
showcase this approach in the case of Fourier imaging, and verify its stability
and performance via a series of numerical experiments. The key impact of this
work is demonstrating the construction of efficient neural networks based on
unrolling with guaranteed stability and accuracy.
