  Subspace clustering is an important unsupervised clustering approach. It is
based on the assumption that the high-dimensional data points are approximately
distributed around several low-dimensional linear subspaces. The majority of
the prominent subspace clustering algorithms rely on the representation of the
data points as linear combinations of other data points, which is known as a
self-expressive representation. To overcome the restrictive linearity
assumption, numerous nonlinear approaches were proposed to extend successful
subspace clustering approaches to data on a union of nonlinear manifolds. In
this comparative study, we provide a comprehensive overview of nonlinear
subspace clustering approaches proposed in the last decade. We introduce a new
taxonomy to classify the state-of-the-art approaches into three categories,
namely locality preserving, kernel based, and neural network based. The major
representative algorithms within each category are extensively compared on
carefully designed synthetic and real-world data sets. The detailed analysis of
these approaches unfolds potential research directions and unsolved challenges
in this field.
