Abstract

An I-by-J-by-K array has rank 1 if the array is the outer product of an I-, a J-, and a K-vector. The authors prove that a three-way array can be uniquely decomposed as the sum of F rank-1 arrays if the F vectors corresponding to two of the ways are linearly independent and the F vectors corresponding to the third way have the property that no two are collinear. Several algorithms that implement the decomposition are described. The algorithms are applied to obtain initial values for nonlinear least-squares calculations. The performances of the decompositions and of the nonlinear least-squares solutions on real and on simulated data are compared. An extension to higher-way arrays is introduced, and the method is compared with those of other authors.

MSC codes

  1. 15A23
  2. 15A69
  3. 62H25
  4. 62J99

Keywords

  1. alternating least-squares algorithm
  2. array rank
  3. multiway arrays

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Information & Authors

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Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1064 - 1083
ISSN (online): 1095-7162

History

Submitted: 10 September 1990
Accepted: 1 January 1992
Published online: 31 July 2006

MSC codes

  1. 15A23
  2. 15A69
  3. 62H25
  4. 62J99

Keywords

  1. alternating least-squares algorithm
  2. array rank
  3. multiway arrays

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