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.