A decomposition for three-way arrays

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.