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Similarly, we may divide into a training and a test set along the sessions, approx-
imate the training set by means of offline learning (recommendable by means of the
online HOSVD from Algorithm 9.2), and use the approximation to forecast the
remaining session entries in the test set.
Of course, we would prefer the efficient way of left-projection approximations,
similarly to Sect. 8.3.3 . First, we have, correspondingly to ( 8.18 ) according to
Definition 9.4,
A t ¼ A 1 U 1 U 1 2 ... d 1 U d 1 U d 1 U d U d ,
ð 9
:
4 Þ
i.e., the approximated matrix is the multilinear product of the projections of A on the
spaces of the singular vector bases of its n -matricizations. Thus, the question is: can
we generalize the left-projection approximation ( 8.20 ) to the d -dimensional case for
the slice B , i.e.,
B t ¼ B 1 U 1 U 1 2 ... d 1 U d 1 U d 1 ?
ð 9 : 5 Þ
Besides the formal analogy, this conjecture is supported by the fact that, in the
HOSVD Algorithm 9.2, too, the projection ( 9.5 ) is carried out at the end. Of course,
the answer is negative (since, otherwise, the frontal mode in Algorithm 9.2 would
be redundant).
Sadly enough, the left-projection approximation is not exact for n
>
2, i.e., not
consistent with ( 9.4 ), since it holds in some cases that
6¼ A 1 U 1 U 1 2 ... d 1 U d 1 U d 1 :
A t
One may easily convince oneself of this by a straightforward evaluation of an
example. Only for the special case that the SVD corresponding to the frontal mode
A ( d ) is of full rank does ( 9.5 ) hold unrestrictedly. This does not mean that ( 9.5 )is
outright useless for the high-dimensional case; in practice, it often yields suffi-
ciently good results, as we shall see later. But caution is advised.
Example 9.3 In the two-dimensional case, the “slice” B corresponds to a vector,
e.g., the products (dimension 1) within a session (dimension 2), and we obtain
according to ( 9.5 )
B t ¼ U 1 U 1 B ,
which complies with the SVD case ( 8.20 ).
In the following, we shall present experimental results for a real-world data set
and compare those to the predictions obtained from a three-dimensional tensor
factorization. To this end, we consider the transaction data of a mail-order
company. The considered data set encompasses 3,016 products and consists of
25,000 transactions from some 800 sessions. We split the data set into a training
set of 20,000 transactions, from which we learn the initial factorization model, and
the actual test set of 5,000 transactions.
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