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arXiv · 1405.7786

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

Namgil Lee·Andrzej Cichocki
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Abstract

We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.

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Namgil Lee, Andrzej Cichocki. 2014-05-30. Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats. https://arxiv.org/abs/1405.7786

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