A DECOMPOSITION FOR 3-WAY ARRAYS

An I-by-J-by-K array has rank 1 if the array is the outer product of a n I-, a J-, and a K-vector. The authors prove that a three-way array c an be uniquely decomposed as the sum of F rank-1 arrays if the F vecto rs 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 no nlinear least-squares calculations. The performances of the decomposit ions and of the nonlinear least-squares solutions on real and on simul ated data are compared. An extension to higher-way arrays is introduce d, and the method is compared with those of other authors.