  In this work we address the problem of solving ill-posed inverse problems in
imaging where the prior is a variational autoencoder (VAE). 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 (JPMAP)
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. We also highlight the importance of correctly training the
VAE using a denoising criterion, in order to ensure that the encoder
generalizes well to out-of-distribution images, without affecting the quality
of the generative model. This simple modification is key to providing
robustness to the whole procedure. Finally we show how our joint MAP
methodology relates to more common MAP approaches, and we propose a
continuation scheme that makes use of our JPMAP algorithm to provide more
robust MAP estimates. 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.
