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

摘要

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.