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.

MSC codes

  1. 15A23
  2. 15A69
  3. 65F99


  1. tensors
  2. high-dimensional problems
  3. SVD
  4. TT-format

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 2295 - 2317
ISSN (online): 1095-7197


Submitted: 10 March 2009
Accepted: 19 June 2011
Published online: 22 September 2011

MSC codes

  1. 15A23
  2. 15A69
  3. 65F99


  1. tensors
  2. high-dimensional problems
  3. SVD
  4. TT-format



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