  We address the problem of compressed sensing using a deep generative prior
model and consider both linear and learned nonlinear sensing mechanisms, where
the nonlinear one involves either a fully connected neural network or a
convolutional neural network. Recently, it has been argued that the
distribution of natural images do not lie in a single manifold but rather lie
in a union of several submanifolds. We propose a sparsity-driven latent space
sampling (SDLSS) framework and develop a proximal meta-learning (PML) algorithm
to enforce sparsity in the latent space. SDLSS allows the range-space of the
generator to be considered as a union-of-submanifolds. We also derive the
sample complexity bounds within the SDLSS framework for the linear measurement
model. The results demonstrate that for a higher degree of compression, the
SDLSS method is more efficient than the state-of-the-art method. We first
consider a comparison between linear and nonlinear sensing mechanisms on
Fashion-MNIST dataset and show that the learned nonlinear version is superior
to the linear one. Subsequent comparisons with the deep compressive sensing
(DCS) framework proposed in the literature are reported. We also consider the
effect of the dimension of the latent space and the sparsity factor in
validating the SDLSS framework. Performance quantification is carried out by
employing three objective metrics: peak signal-to-noise ratio (PSNR),
structural similarity index metric (SSIM), and reconstruction error (RE).
