  Many problems in computational neuroscience, neuroinformatics, pattern/image
recognition, signal processing and machine learning generate massive amounts of
multidimensional data with multiple aspects and high dimensionality. Tensors
(i.e., multi-way arrays) provide often a natural and compact representation for
such massive multidimensional data via suitable low-rank approximations. Big
data analytics require novel technologies to efficiently process huge datasets
within tolerable elapsed times. Such a new emerging technology for
multidimensional big data is a multiway analysis via tensor networks (TNs) and
tensor decompositions (TDs) which represent tensors by sets of factor
(component) matrices and lower-order (core) tensors. Dynamic tensor analysis
allows us to discover meaningful hidden structures of complex data and to
perform generalizations by capturing multi-linear and multi-aspect
relationships. We will discuss some fundamental TN models, their mathematical
and graphical descriptions and associated learning algorithms for large-scale
TDs and TNs, with many potential applications including: Anomaly detection,
feature extraction, classification, cluster analysis, data fusion and
integration, pattern recognition, predictive modeling, regression, time series
analysis and multiway component analysis.
  Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models,
Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA),
Tensorization/Quantization, tensor train (TT/QTT) - Matrix Product States
(MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP).
