  Deep learning has gained great popularity due to its widespread success on
many inference problems. We consider the application of deep learning to the
sparse linear inverse problem, where one seeks to recover a sparse signal from
a few noisy linear measurements. In this paper, we propose two novel
neural-network architectures that decouple prediction errors across layers in
the same way that the approximate message passing (AMP) algorithms decouple
them across iterations: through Onsager correction. First, we propose a
"learned AMP" network that significantly improves upon Gregor and LeCun's
"learned ISTA." Second, inspired by the recently proposed "vector AMP" (VAMP)
algorithm, we propose a "learned VAMP" network that offers increased robustness
to deviations in the measurement matrix from i.i.d. Gaussian. In both cases, we
jointly learn the linear transforms and scalar nonlinearities of the network.
Interestingly, with i.i.d. signals, the linear transforms and scalar
nonlinearities prescribed by the VAMP algorithm coincide with the values
learned through back-propagation, leading to an intuitive interpretation of
learned VAMP. Finally, we apply our methods to two problems from 5G wireless
communications: compressive random access and massive-MIMO channel estimation.
