A alternative notation for the PARAFAC model

Using the tensor products notation (see Chapter 2) an alternative expression of the PARAFAC model is  

The PARAFAC model can also be written by using the vec-operator and the Kronecker product. This is given by 

A final way of writing the PARAFAC model is given by the Khatri-Rao product. 

It is instructive to have an understanding of why the Khatri-Rao product is useful for multi-linear models. Starting with a bilinear two-way model 

every bilinear component vec(arb7) can be written as a well-defined vector that is a function of ar and br only. This vector is the Kronecker product of br and ar. Define the Khatri-Rao product of two matrices, A and B, with the same number of columns as 

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