A weak degeneracy revealing decomposition for the CANDECOMP/PARAFAC model

The CANDECOMP/PARAFAC (CP) model is a well known and frequently used tool for extracting substantial information from a three-way data array. It has several useful characteristics and usually gives meaningful insights about the underlying structure of the data. However, in some cases it has a ‘strange’ behaviour suffering from the so-called ‘degenerate solutions’, i.e. solutions where the components show a diverging pattern and are meaningless. Several authors have investigated the causes of degeneracy concluding that the phenomenon is due to a lack of minimum of the loss function. In this paper, we study the degeneracy of CP limiting our attention to the two-component case. The study is done by introducing a canonical form, called 2DR, which is ‘weakly degeneracy revealing’. On the ground of this framework, degeneracy is studied along with some of the remedies proposed in the literature by using a Tucker3 model having a core in the 2DR form. The analysis gives new insights about the behaviour of the CP model and suggests new ideas on how to deal with degeneracy.