Solving Inverse Problems by Joint Posterior Maximization with a VAE Prior

TL;DR

This paper introduces JPMAP, a novel method for solving ill-posed inverse imaging problems using a joint posterior maximization approach with a VAE prior, offering better convergence and solution quality.

Contribution

It proposes a new joint posterior maximization framework that leverages a VAE prior, enabling efficient optimization and improved results over existing non-convex MAP methods.

Findings

JPMAP converges to stationary points due to weak bi-convexity.

Experimental results show JPMAP outperforms other non-convex MAP approaches.

The method is effective across various degradation models without retraining the prior.

Abstract

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.