  The widespread use of multisensor technology and the emergence of big data
sets have brought the necessity to develop more versatile tools to represent
higher-order data with multiple aspects and high dimensionality. Data in the
form of multidimensional arrays, also referred to as tensors, arises in a
variety of applications including chemometrics, hyperspectral imaging, high
resolution videos, neuroimaging, biometrics, and social network analysis. Early
multiway data analysis approaches reformatted such tensor data as large vectors
or matrices and then resorted to dimensionality reduction methods developed for
classical two-way analysis such as PCA. However, one cannot discover hidden
components within multiway data using conventional PCA. To this end, tensor
decomposition methods which are flexible in the choice of the constraints and
that extract more general latent components have been proposed. In this paper,
we review the major tensor decomposition methods with a focus on problems
targeted by classical PCA. In particular, we present tensor methods that aim to
solve three important challenges typically addressed by PCA: dimensionality
reduction, i.e. low-rank tensor approximation, supervised learning, i.e.
learning linear subspaces for feature extraction, and robust low-rank tensor
recovery. We also provide experimental results to compare different tensor
models for both dimensionality reduction and supervised learning applications.
