Cramér-Rao lower bounds for low-rank decomposition of multidimensional arrays

Xiangqian Liu, Nikolaos Sidiropoulos

Research output: Contribution to journalArticlepeer-review

161 Scopus citations


Unlike low-rank matrix decomposition, which is generically nonunique for rank greater than one, low-rank three-and higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher dimensional arrays. This paper develops Cramér-Rao Bound (CRB) results for low-rank decomposition of three-and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided.

Original languageEnglish (US)
Pages (from-to)2074-2086
Number of pages13
JournalIEEE Transactions on Signal Processing
Issue number9
StatePublished - Sep 2001

Bibliographical note

Funding Information:
Manuscript received July 17, 2000; revised May 31, 2001. This work was supported by the National Science Foundation under CAREER grant 0096165 and Wireless grant 0096164. The associate editor coordinating the review of this paper and approving it for publication was Dr. Brian Sadler.


  • Cramér-Rao bound
  • Least squares method
  • Matrix decomposition
  • Multidimensional signal processing


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