, Volume 65, Issue 4, pp 525–532 | Cite as

The typical rank of tall three-way arrays

  • Jos M. F. ten Berge


The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J.

Key words

three-way rank tensorial rank CANDECOMP PARAFAC three-way component analysis 


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Copyright information

© The Psychometric Society 2000

Authors and Affiliations

  1. 1.Heijmans Institute of Psychological ResearchUniversity of GroningenGroningenThe Netherlands

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