Learning Posterior Distributions in Underdetermined Inverse Problems

Abstract

In recent years, classical knowledge-driven approaches for inverse problems have been complemented by data-driven methods exploiting the power of machine and especially deep learning. Purely data-driven methods, however, come with the drawback of disregarding prior knowledge of the problem even though it has shown to be beneficial to incorporate this knowledge into the problem-solving process. We thus introduce an unpaired learning approach for learning posterior distributions of underdetermined inverse problems. It combines advantages of deep generative modeling with established ideas of knowledge-driven approaches by incorporating prior information about the inverse problem. We develop a new neural network architecture ’UnDimFlow’ (short for Unequal Dimensionality Flow) consisting of two normalizing flows, one from the data to the latent, and one from the latent to the solution space. Additionally, we incorporate the forward operator to develop an unpaired learning method for the UnDimFlow architecture and propose a tailored point estimator to recover an optimal solution during inference. We evaluate our method on the two underdetermined inverse problems of image inpainting and super-resolution.