Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior

TL;DR

This paper introduces JPMAP, a novel method for solving ill-posed imaging inverse problems using a joint posterior maximization approach with a variational autoencoder prior, ensuring convergence and robustness.

Contribution

The paper proposes JPMAP, a joint MAP optimization method with theoretical convergence guarantees and a robust training strategy for VAEs, improving solution quality in inverse imaging problems.

Findings

JPMAP converges to stationary points under weak bi-convexity.

Training VAEs with a denoising criterion improves out-of-distribution generalization.

JPMAP outperforms other non-convex MAP methods in solution quality.

Abstract

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.