Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This chapter provides a unified framework to handle these approaches via Markov chains. We 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 us to couple 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. Our framework establishes a useful mathematical tool to combine the various approaches.

Deep generative models for approximating complicated and often high-dimensional probability distributions became a rapidly developing research field. Normalizing flows are a popular subclass of these generative models. They can be used to model a target distribution by a simpler latent distribution which is usually the standard normal distribution. In this paper, we are interested in finite normalizing flows which are basically concatenations of learned diffeomorphisms. The parameters of the diffeomorphism are adapted to the target distribution by minimizing a loss functions. To this end, the diffeomorphism must have a tractable Jacobian determinant. For the continuous counterpart of normalizing flows, we refer to the overview paper [43] and the references therein. Suitable architectures of finite normalizing flows include invertible residual neural networks (ResNets) [7, 11, 22], (coupling-based) invertible neural networks (INNs) [4, 14, 29, 34, 40] and autoregessive flows [13, 15, 26, 38].