  Constrained low-rank matrix approximations have been known for decades as
powerful linear dimensionality reduction techniques to be able to extract the
information contained in large data sets in a relevant way. However, such
low-rank approaches are unable to mine complex, interleaved features that
underlie hierarchical semantics. Recently, deep matrix factorization (deep MF)
was introduced to deal with the extraction of several layers of features and
has been shown to reach outstanding performances on unsupervised tasks. Deep MF
was motivated by the success of deep learning, as it is conceptually close to
some neural networks paradigms. In this paper, we present the main models,
algorithms, and applications of deep MF through a comprehensive literature
review. We also discuss theoretical questions and perspectives of research.
