  Tensors are multidimensional arrays of numerical values and therefore
generalize matrices to multiple dimensions. While tensors first emerged in the
psychometrics community in the $20^{\text{th}}$ century, they have since then
spread to numerous other disciplines, including machine learning. Tensors and
their decompositions are especially beneficial in unsupervised learning
settings, but are gaining popularity in other sub-disciplines like temporal and
multi-relational data analysis, too.
  The scope of this paper is to give a broad overview of tensors, their
decompositions, and how they are used in machine learning. As part of this, we
are going to introduce basic tensor concepts, discuss why tensors can be
considered more rigid than matrices with respect to the uniqueness of their
decomposition, explain the most important factorization algorithms and their
properties, provide concrete examples of tensor decomposition applications in
machine learning, conduct a case study on tensor-based estimation of mixture
models, talk about the current state of research, and provide references to
available software libraries.
