Cramér-Rao-induced bounds for CANDECOMP/PARAFAC tensor decomposition

Petr Tichavský, Anh Huy Phan, Zbyněk Koldovský

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32 Citations (Scopus)


This paper presents a Cramér-Rao lower bound (CRLB) on the variance of unbiased estimates of factor matrices in Canonical Polyadic (CP) or CANDECOMP/PARAFAC (CP) decompositions of a tensor from noisy observations, (i.e., the tensor plus a random Gaussian i.i.d. tensor). A novel expression is derived for a bound on the mean square angular error of factors along a selected dimension of a tensor of an arbitrary dimension. The expression needs less operations for computing the bound, O(NR6), than the best existing state-of-the art algorithm, O(N3R6) operations, where N and R are the tensor order and the tensor rank. Insightful expressions are derived for tensors of rank 1 and rank 2 of arbitrary dimension and for tensors of arbitrary dimension and rank, where two factor matrices have orthogonal columns.

Original languageEnglish
Article number6457481
Pages (from-to)1986-1997
Number of pages12
JournalIEEE Transactions on Signal Processing
Issue number8
Publication statusPublished - 2013
Externally publishedYes


  • Canonical polyadic decomposition
  • Cramér-Rao lower bound
  • multilinear models
  • stability
  • uniqueness


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