Rank of three-way arrays

Although there is a clear definition of the rank of three-way arrays, the properties of two-way rank do not automatically generalize to three-way rank. This will be shown in the following 

The rank of a three-way array X is the smallest number of trilinear (PARAFAC) components needed to fit X exactly. This coincides with the definition of two-way rank it is a direct 

Consider all arrays of size 2x2x2. Then it can be shown that the maximum possible rank of such an array is 3 [Kruskal 1989], This is in remarkable contrast with two-way arrays a 2 x 2 array (matrix) has maximum rank 2. Likewise, it has been shown that the maximum rank of a 3 x 3 x 3 array is 5 [Kruskal 1989]. Results are also available for the maximum rank of a 2 x J x K array [Ja Ja 1979] which is m + min(m,n), where m = m n(J,K). n = [max(./,A )/2] and [x] means the largest integer x. 

Kruskalhasshownthatthereisapositiveprobability3 forarandomlygenerated2 x 2 x 2 array to have a rank lower than three [Kruskal 1989], In a simulation study he found that of all 2 x 2 x 2 arrays, there is a probability of 0.79 of obtaining a rank two array and a probability of 0.21 of obtaining a rank three array. The probability of obtaining an array of rank one or lower is zero. These results have been generalized to the case of 2 xn x n arrays [Ten Berge 1991], This is in sharp contrast with results for two-way arrays, where the full rank case has probability one. 

Due to the special properties of three-way rank, it is important to distinguish different types of rank for three-way arrays [Kruskal 1989, Lickteig 1985]  

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The pseudo-rank of three-way arrays is a straightforward generalization of the two-way definition. Suppose that a three-way array X is modeled with a three-way model and this results in... 

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