Two-factor degeneracies and a stabilization of PARAFAC

Mitchell and Burdick (B.C. Mitchell, D.S. Burdick, An empirical comparison of resolution methods for three-way arrays, Chemom. Intell. Lab. Syst. 20 (1993) 149–161; B.C. Mitchell, D.S. Burdick, Slowly converging PARAFAC sequences: Swamps and two-factor degeneracies, J. Chemom. 8 (1994) 155–168.) uncovered an intriguing correspondence between the existence of PARAFAC swamps and the presence of two-factor degeneracies (2FDs). This observation, coupled with the recognition that post-swamp resolutions were generally better than pre-swamp resolutions allowed the user a method of detecting when a swamp had likely been encountered and, hence, when it was safe to assume that PARAFAC had converged to a suitable resolution. Still, this correspondence alone did not suggest how one might reduce the number of PARAFAC iterations required to move through a swamp. In this paper, it is noted that serious 2FDs must produce an identifiable ill-conditioning in the least squares estimation step in PARAFAC. Moreover, a serious 2FD is only one way this ill-conditioning may occur and, hence, it is more correct to say that swamps correspond to this ill-conditioning in general, rather than to the presence of 2FDs in particular. In an attempt to reduce the number of iterations that PARAFAC spends in a swamp, a particular method of stabilization is employed and results are presented which suggest that the number iterations can often be greatly reduced.