Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This Element provides a unified framework to handle these approaches via Markov chains. The authors consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties, and show how many state-of-the-art models for data generation fit into this framework. Indeed numerical simulations show that including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables the coupling of both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers, variational autoencoders and diffusion normalizing flows in a mathematically sound way. The authors' framework establishes a useful mathematical tool to combine the various approaches.
Learning neural networks using only a small amount of data is an important ongoing research topic with tremendous potential for applications. In this paper, we introduce a regularizer for the variational modeling of inverse problems in imaging based on normalizing flows. Our regularizer, called patchNR, involves a normalizing flow learned on patches of very few images. The subsequent reconstruction method is completely unsupervised and the same regularizer can be used for different forward operators acting on the same class of images. By investigating the distribution of patches versus those of the whole image class, we prove that our variational model is indeed a MAP approach. Our model can be generalized to conditional patchNRs, if additional supervised information is available. Numerical examples for low-dose CT, limited-angle CT and superresolution of material images demonstrate that our method provides high quality results among unsupervised methods, but requires only few data.
In this paper, we analyze the properties of invertible neural networks, which provide a way of solving inverse problems. Our main focus lies on investigating and controlling the Lipschitz constants of the corresponding inverse networks. Without such a control, numerical simulations are prone to errors and not much is gained against traditional approaches. Fortunately, our analysis indicates that changing the latent distribution from a standard normal one to a Gaussian mixture model resolves the issue of exploding Lipschitz constants. Indeed, numerical simulations confirm that this modification leads to significantly improved sampling quality in multimodal applications.
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