  A central problem in unsupervised deep learning is how to find useful
representations of high-dimensional data, sometimes called "disentanglement".
Most approaches are heuristic and lack a proper theoretical foundation. In
linear representation learning, independent component analysis (ICA) has been
successful in many applications areas, and it is principled, i.e., based on a
well-defined probabilistic model. However, extension of ICA to the nonlinear
case has been problematic due to the lack of identifiability, i.e., uniqueness
of the representation. Recently, nonlinear extensions that utilize temporal
structure or some auxiliary information have been proposed. Such models are in
fact identifiable, and consequently, an increasing number of algorithms have
been developed. In particular, some self-supervised algorithms can be shown to
estimate nonlinear ICA, even though they have initially been proposed from
heuristic perspectives. This paper reviews the state-of-the-art of nonlinear
ICA theory and algorithms.
