  The promise of compressive sensing (CS) has been offset by two significant
challenges. First, real-world data is not exactly sparse in a fixed basis.
Second, current high-performance recovery algorithms are slow to converge,
which limits CS to either non-real-time applications or scenarios where massive
back-end computing is available. In this paper, we attack both of these
challenges head-on by developing a new signal recovery framework we call {\em
DeepInverse} that learns the inverse transformation from measurement vectors to
signals using a {\em deep convolutional network}. When trained on a set of
representative images, the network learns both a representation for the signals
(addressing challenge one) and an inverse map approximating a greedy or convex
recovery algorithm (addressing challenge two). Our experiments indicate that
the DeepInverse network closely approximates the solution produced by
state-of-the-art CS recovery algorithms yet is hundreds of times faster in run
time. The tradeoff for the ultrafast run time is a computationally intensive,
off-line training procedure typical to deep networks. However, the training
needs to be completed only once, which makes the approach attractive for a host
of sparse recovery problems.
