Improving PARAFAC-ALS performance by initialization

The CANDECOMP/PARAFAC (CP) model (Carroll and Chang, 1970; Harshman, 1970) is a trilinear decomposition which provides a low rank approximation of a three-way array in a manner that preserves the multi-mode structure of the data. This is achieved by estimating three sets of parameters, one for each dimension of the array, namely observation units, variables and occasions. The CP model, however, due to an elevated number of degrees of freedom, can be quite challenging to estimate. The most commonly used algorithm to t this model to the data is PARAFAC-ALS. Comparative studies (Tomasi and Bro, 2006) have shown that this procedure is, in general, more reliable and accurate than other algorithms proposed in the literature. Nonetheless, it presents some non-trivial issues: it can be slow at converging and may run into over-factoring and bad initialization degeneracies. With respect to these setbacks, some of the alternative estimating procedures are able to perform better than ALS, specically the Alternating Trilinear Decomposition (ATLD) and Self-weighted Alternating Trilin-ear Decomposition (SWATLD) proposed by Wu et al. (1998) and Chen et al. (2000) respectively. These algorithms are faster and less likely to be aected by over-factoring and bad initial values. They present, however, diculties connected to their non-least squares objective functions and for this reason they are seldom used in practice. In this work it is suggested that a successful way to improve on ALS performance with respect to the presented drawbacks is to initialize it with either ATLD or SWATLD steps, obtaining two integrated ALS procedures. The eectiveness of this methodology is demonstrated by comparing the results of standard ALS with the ones of the proposed integrated ALS variants in an extensive simulation design.