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Machine learning and data mining algorithms are becoming increasinglyimportant in analyzing large volume, multi-relational and multi--modaldatasets, which are often conveniently represented as multiway arrays ortensors. It is therefore timely and valuable for the multidisciplinary researchcommunity to review tensor decompositions and tensor networks as emerging toolsfor large-scale data analysis and data mining. We provide the mathematical andgraphical representations and interpretation of tensor networks, with the mainfocus on the Tucker and Tensor Train (TT) decompositions and their extensionsor generalizations.  Keywords: Tensor networks, Function-related tensors, CP decomposition, Tuckermodels, tensor train (TT) decompositions, matrix product states (MPS), matrixproduct operators (MPO), basic tensor operations, multiway component analysis,multilinear blind source separation, tensor completion, linear/multilineardimensionality reduction, large-scale optimization problems, symmetriceigenvalue 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)

Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1