  Machine learning and data mining algorithms are becoming increasingly
important in analyzing large volume, multi-relational and multi--modal
datasets, which are often conveniently represented as multiway arrays or
tensors. It is therefore timely and valuable for the multidisciplinary research
community to review tensor decompositions and tensor networks as emerging tools
for large-scale data analysis and data mining. We provide the mathematical and
graphical representations and interpretation of tensor networks, with the main
focus on the Tucker and Tensor Train (TT) decompositions and their extensions
or generalizations.
  Keywords: Tensor networks, Function-related tensors, CP decomposition, Tucker
models, tensor train (TT) decompositions, matrix product states (MPS), matrix
product operators (MPO), basic tensor operations, multiway component analysis,
multilinear blind source separation, tensor completion, linear/multilinear
dimensionality reduction, large-scale optimization problems, symmetric
eigenvalue decomposition (EVD), PCA/SVD, huge systems of linear equations,
pseudo-inverse of very large matrices, Lasso and Canonical Correlation Analysis
(CCA) (This is Part 1)
