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In this work we address the problem of solving ill-posed inverse problems inimaging where the prior is a variational autoencoder (VAE). Specifically weconsider the decoupled case where the prior is trained once and can be reusedfor many different log-concave degradation models without retraining. Whereasprevious MAP-based approaches to this problem lead to highly non-convexoptimization algorithms, our approach computes the joint (space-latent) MAPthat naturally leads to alternate optimization algorithms and to the use of astochastic encoder to accelerate computations. The resulting technique (JPMAP)performs Joint Posterior Maximization using an Autoencoding Prior. We showtheoretical and experimental evidence that the proposed objective function isquite close to bi-convex. Indeed it satisfies a weak bi-convexity propertywhich is sufficient to guarantee that our optimization scheme converges to astationary point. We also highlight the importance of correctly training theVAE using a denoising criterion, in order to ensure that the encodergeneralizes well to out-of-distribution images, without affecting the qualityof the generative model. This simple modification is key to providingrobustness to the whole procedure. Finally we show how our joint MAPmethodology relates to more common MAP approaches, and we propose acontinuation scheme that makes use of our JPMAP algorithm to provide morerobust MAP estimates. Experimental results also show the higher quality of thesolutions obtained by our JPMAP approach with respect to other non-convex MAPapproaches which more often get stuck in spurious local optima.

Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior