  We study a deep learning inspired formulation for the blind demodulation
problem, which is the task of recovering two unknown vectors from their
entrywise multiplication. We consider the case where the unknown vectors are in
the range of known deep generative models,
$\mathcal{G}^{(1)}:\mathbb{R}^n\rightarrow\mathbb{R}^\ell$ and
$\mathcal{G}^{(2)}:\mathbb{R}^p\rightarrow\mathbb{R}^\ell$. In the case when
the networks corresponding to the generative models are expansive, the weight
matrices are random and the dimension of the unknown vectors satisfy $\ell =
\Omega(n^2+p^2)$, up to log factors, we show that the empirical risk objective
has a favorable landscape for optimization. That is, the objective function has
a descent direction at every point outside of a small neighborhood around four
hyperbolic curves. We also characterize the local maximizers of the empirical
risk objective and, hence, show that there does not exist any other stationary
points outside of these neighborhood around four hyperbolic curves and the set
of local maximizers. We also implement a gradient descent scheme inspired by
the geometry of the landscape of the objective function. In order to converge
to a global minimizer, this gradient descent scheme exploits the fact that
exactly one of the hyperbolic curve corresponds to the global minimizer, and
thus points near this hyperbolic curve have a lower objective value than points
close to the other spurious hyperbolic curves. We show that this gradient
descent scheme can effectively remove distortions synthetically introduced to
the MNIST dataset.
