  Compressive image recovery is a challenging problem that requires fast and
accurate algorithms. Recently, neural networks have been applied to this
problem with promising results. By exploiting massively parallel GPU processing
architectures and oodles of training data, they can run orders of magnitude
faster than existing techniques. However, these methods are largely
unprincipled black boxes that are difficult to train and often-times specific
to a single measurement matrix.
  It was recently demonstrated that iterative sparse-signal-recovery algorithms
can be "unrolled" to form interpretable deep networks. Taking inspiration from
this work, we develop a novel neural network architecture that mimics the
behavior of the denoising-based approximate message passing (D-AMP) algorithm.
We call this new network Learned D-AMP (LDAMP).
  The LDAMP network is easy to train, can be applied to a variety of different
measurement matrices, and comes with a state-evolution heuristic that
accurately predicts its performance. Most importantly, it outperforms the
state-of-the-art BM3D-AMP and NLR-CS algorithms in terms of both accuracy and
run time. At high resolutions, and when used with sensing matrices that have
fast implementations, LDAMP runs over $50\times$ faster than BM3D-AMP and
hundreds of times faster than NLR-CS.
