  We examine the theoretical properties of enforcing priors provided by
generative deep neural networks via empirical risk minimization. In particular
we consider two models, one in which the task is to invert a generative neural
network given access to its last layer and another in which the task is to
invert a generative neural network given only compressive linear observations
of its last layer. We establish that in both cases, in suitable regimes of
network layer sizes and a randomness assumption on the network weights, that
the non-convex objective function given by empirical risk minimization does not
have any spurious stationary points. That is, we establish that with high
probability, at any point away from small neighborhoods around two scalar
multiples of the desired solution, there is a descent direction. Hence, there
are no local minima, saddle points, or other stationary points outside these
neighborhoods. These results constitute the first theoretical guarantees which
establish the favorable global geometry of these non-convex optimization
problems, and they bridge the gap between the empirical success of enforcing
deep generative priors and a rigorous understanding of non-linear inverse
problems.
