Degeneracy in Candecomp/Parafac explained for p x p x 2 arrays of rank p+1 or higher

The Candecomp/Parafac (CP) model decomposes a three-way array into a prespecified number R of rank-1 arrays and a residual array, in which the sum of squares of the residual array is minimized. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate solutions, in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large. We consider the real-valued CP model in which p x p x 2 arrays of rank p + 1 or higher are decomposed into p rank-1 arrays and a residual array. It is shown that the CP objective function does not have a minimum in these cases, but an infimum. Moreover, any sequence of CP approximations, of which the objective value approaches the infimum, will become degenerate. This result extends Ten Berge, Kiers, & De Leeuw (1988), who consider a particular 2 x 2 x 2 array of rank 3.