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Then its matricizations are given by
,
1100
1100
A
ðÞ
¼
,
1100
1100
A
ðÞ
¼
and
1111
0000
A
ðÞ
¼
:
■
9.1.2 And Why We Should Care
In classical data mining, matrices are deployed to model weighted binary relations,
such as the session-product relation in the framework presented in Sect.
8.1
. Tensor
algebra provides a means to model higher-order relations. The latter, e.g., arise in
context-aware models: with regard to user-rating prediction, it is natural to expect
that certain extrinsic circumstances of the situation in which a user is prompted for a
rating affect the latter. As an illustration, consider the following datasets described
by [KABO10]:
[The first dataset] contains 1464 ratings by 84 users for 192 movies.
[The users] were asked
to fill out a questionnaire on movies using a rating scale ranging from 1
...
.They
were also queried on additional information about the context of the movie-watching expe-
rience[:]
...
to 13
...
...
companion, day of the week, if it was on the opening weekend, season, and year
seen.
...
[The second dataset] contains food rating data from 212 users on 20 food menus.
The users
were asked to rate the food menu while being in different levels of hunger. Moreover, some
ratings were done when really experiencing the situation (i.e., participants were hungry and
ordered the menu) and some while imagining the situation. For example, in the virtual
situation participants could be full, but should have provided a rating for a food menu
imagining that they are hungry.
The above-outlined datasets are established upon relations of genuinely high order.
Beyond that, it is also possible to construct higher-order relations by combining
lower-order ones. For example, users may endow products in a shop with tags and
provide (implicit or explicit) ratings for both products and tags. This gives rise to
two weighted relations, which may be combined as follows: if a particular user
u
endows the product
p
with a rating score
S
p
which is tagged with
t
, which, in turn,
has been given a score of
s
t
by the user
u
, then the triplet (
u,s,t
) is assigned with the
value
s
p
s
t
. In a similar fashion, one might take background information on users
and products into account, which leads to relations of arbitrary order.
The above examples give rise to the question of whether the previously
developed framework of PCA-based CF may be extended to the tensor case in a
...
