  Deep generative models have emerged as a powerful class of priors for signals
in various inverse problems such as compressed sensing, phase retrieval and
super-resolution. Here, we assume an unknown signal to lie in the range of some
pre-trained generative model. A popular approach for signal recovery is via
gradient descent in the low-dimensional latent space. While gradient descent
has achieved good empirical performance, its theoretical behavior is not well
understood. In this paper, we introduce the use of stochastic gradient Langevin
dynamics (SGLD) for compressed sensing with a generative prior. Under mild
assumptions on the generative model, we prove the convergence of SGLD to the
true signal. We also demonstrate competitive empirical performance to standard
gradient descent.
