NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems

• URL: http://arxiv.org/abs/2304.08342v2
• Date: Sat, 14 Oct 2023 16:46:08 GMT
• Title: NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems
• Authors: Ziruo Cai, Junqi Tang, Subhadip Mukherjee, Jinglai Li, Carola Bibiane Sch\"onlieb, Xiaoqun Zhang
• Abstract summary: We introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.
• Score: 7.38079566297881
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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