  Normalizing Flows (NFs) describe a class of models that express a complex
target distribution as the composition of a series of bijective transformations
over a simpler base distribution. By limiting the space of candidate
transformations to diffeomorphisms, NFs enjoy efficient, exact sampling and
density evaluation, enabling NFs to flexibly behave as both discriminative and
generative models. Their restriction to diffeomorphisms, however, enforces that
input, output and all intermediary spaces share the same dimension, limiting
their ability to effectively represent target distributions with complex
topologies. Additionally, in cases where the prior and target distributions are
not homeomorphic, Normalizing Flows can leak mass outside of the support of the
target. This survey covers a selection of recent works that combine aspects of
other generative model classes, such as VAEs and score-based diffusion, and in
doing so loosen the strict bijectivity constraints of NFs to achieve a balance
of expressivity, training speed, sample efficiency and likelihood tractability.

Machine Learning;Artificial Intelligence