  Bayesian methods for solving inverse problems are a powerful alternative to
classical methods since the Bayesian approach offers the ability to quantify
the uncertainty in the solution. In recent years, data-driven techniques for
solving inverse problems have also been remarkably successful, due to their
superior representation ability. In this work, we incorporate data-based models
into a class of Langevin-based sampling algorithms for Bayesian inference in
imaging inverse problems. In particular, we introduce NF-ULA (Normalizing
Flow-based Unadjusted Langevin algorithm), which involves learning a
normalizing flow (NF) as the image prior. We use NF to learn the prior because
a tractable closed-form expression for the log prior enables the
differentiation of it using autograd libraries. Our algorithm only requires a
normalizing flow-based generative network, which can be pre-trained
independently of the considered inverse problem and the forward operator. We
perform theoretical analysis by investigating the well-posedness and
non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of
the proposed NF-ULA algorithm is demonstrated in various image restoration
problems such as image deblurring, image inpainting, and limited-angle X-ray
computed tomography (CT) reconstruction. NF-ULA is found to perform better than
competing methods for severely ill-posed inverse problems.
