  In recent years, Generative Adversarial Networks (GANs) have shown
substantial progress in modeling complex distributions of data. These networks
have received tremendous attention since they can generate implicit
probabilistic models that produce realistic data using a stochastic procedure.
While such models have proven highly effective in diverse scenarios, they
require a large set of fully-observed training samples. In many applications
access to such samples are difficult or even impractical and only noisy or
partial observations of the desired distribution is available. Recent research
has tried to address the problem of incompletely observed samples to recover
the distribution of the data. \citep{zhu2017unpaired} and
\citep{yeh2016semantic} proposed methods to solve ill-posed inverse problem
using cycle-consistency and latent-space mappings in adversarial networks,
respectively. \citep{bora2017compressed} and \citep{kabkab2018task} have
applied similar adversarial approaches to the problem of compressed sensing. In
this work, we focus on a new variant of GAN models called AmbientGAN, which
incorporates a measurement process (e.g. adding noise, data removal and
projection) into the GAN training. While in the standard GAN, the discriminator
distinguishes a generated image from a real image, in AmbientGAN model the
discriminator has to separate a real measurement from a simulated measurement
of a generated image. The results shown by \citep{bora2018ambientgan} are quite
promising for the problem of incomplete data, and have potentially important
implications for generative approaches to compressed sensing and ill-posed
problems.
