  In this paper we address the problem of solving ill-posed inverse problems in
imaging where the prior is a neural generative model. Specifically we consider
the decoupled case where the prior is trained once and can be reused for many
different log-concave degradation models without retraining. Whereas previous
MAP-based approaches to this problem lead to highly non-convex optimization
algorithms, our approach computes the joint (space-latent) MAP that naturally
leads to alternate optimization algorithms and to the use of a stochastic
encoder to accelerate computations. The resulting technique is called JPMAP
because it performs Joint Posterior Maximization using an Autoencoding Prior.
We show theoretical and experimental evidence that the proposed objective
function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity
property which is sufficient to guarantee that our optimization scheme
converges to a stationary point.
  Experimental results also show the higher quality of the solutions obtained
by our JPMAP approach with respect to other non-convex MAP approaches which
more often get stuck in spurious local optima.
