  A Generative Adversarial Network (GAN) with generator $G$ trained to model
the prior of images has been shown to perform better than sparsity-based
regularizers in ill-posed inverse problems. Here, we propose a new method of
deploying a GAN-based prior to solve linear inverse problems using projected
gradient descent (PGD). Our method learns a network-based projector for use in
the PGD algorithm, eliminating expensive computation of the Jacobian of $G$.
Experiments show that our approach provides a speed-up of $60\text{-}80\times$
over earlier GAN-based recovery methods along with better accuracy. Our main
theoretical result is that if the measurement matrix is moderately conditioned
on the manifold range($G$) and the projector is $\delta$-approximate, then the
algorithm is guaranteed to reach $O(\delta)$ reconstruction error in
$O(log(1/\delta))$ steps in the low noise regime. Additionally, we propose a
fast method to design such measurement matrices for a given $G$. Extensive
experiments demonstrate the efficacy of this method by requiring
$5\text{-}10\times$ fewer measurements than random Gaussian measurement
matrices for comparable recovery performance. Because the learning of the GAN
and projector is decoupled from the measurement operator, our GAN-based
projector and recovery algorithm are applicable without retraining to all
linear inverse problems, as confirmed by experiments on compressed sensing,
super-resolution, and inpainting.
