Tensor-Train Decomposition

A simple nonrecursive form of the tensor decomposition in $d$ dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approxi...

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Bibliographic Details
Published inSIAM journal on scientific computing Vol. 33; no. 5; pp. 2295 - 2317
Main Author Oseledets, I. V.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2011
Subjects
Algorithms
Applied mathematics
Approximation
Computational mathematics
Decomposition
Experiments
Linear algebra
Numerical analysis
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Summary:A simple nonrecursive form of the tensor decomposition in $d$ dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.
ISSN:1064-8275
1095-7197
DOI:10.1137/090752286