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We discuss extended definitions of linear and multilinear operations such asKronecker, Hadamard, and contracted products, and establish links between themfor tensor calculus. Then we introduce effective low-rank tensor approximationtechniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT)decompositions with a number of mathematical and graphical representations. Wealso provide a brief review of mathematical properties of the TT decompositionas a low-rank approximation technique. With the aim of breaking thecurse-of-dimensionality in large-scale numerical analysis, we describe basicoperations on large-scale vectors, matrices, and high-order tensors representedby TT decomposition. The proposed representations can be used for describingnumerical methods based on TT decomposition for solving large-scaleoptimization problems such as systems of linear equations and symmetriceigenvalue problems.

Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats