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CHAPTER 9. RECOMMENDATION SYSTEMS
HP1
HP2
HP3
TW
SW1
SW2
SW3
A
2/3
5/3
−7/3
B
1/3
1/3
−2/3
C
−5/3
1/3
4/3
D
0
0
Figure 9.6: The utility matrix introduced in Fig. 9.1
The cosine of the angle between between A and C is
(5/3) × (−5/3) + (−7/3) × (1/3)
p
(2/3)2 + (5/3)2 + (−7/3)2p
(−5/3)2 + (1/3)2 + (4/3)2 = −0.559
Notice that under this measure, A and C are much further apart than A and
B, and neither pair is very close. Both these observations make intuitive sense,
given that A and C disagree on the two movies they rated in common, while A
and B give similar scores to the one movie they rated in common.
✷
9.3.2
The Duality of Similarity
The utility matrix can be viewed as telling us about users or about items, or
both.
It is important to realize that any of the techniques we suggested in
Section 9.3.1 for ﬁnding similar users can be used on columns of the utility
matrix to ﬁnd similar items. There are two ways in which the symmetry is
broken in practice.
1. We can use information about users to recommend items. That is, given
a user, we can ﬁnd some number of the most similar users, perhaps using
the techniques of Chapter 3. We can base our recommendation on the
decisions made by these similar users, e.g., recommend the items that the
greatest number of them have purchased or rated highly. However, there
is no symmetry. Even if we ﬁnd pairs of similar items, we need to take
an additional step in order to recommend items to users. This point is
explored further at the end of this subsection.
2. There is a diﬀerence in the typical behavior of users and items, as it
pertains to similarity. Intuitively, items tend to be classiﬁable in simple
terms. For example, music tends to belong to a single genre. It is impossi-
ble, e.g., for a piece of music to be both 60’s rock and 1700’s baroque. On
the other hand, there are individuals who like both 60’s rock and 1700’s
baroque, and who buy examples of both types of music. The consequence
is that it is easier to discover items that are similar because they belong
to the same genre, than it is to detect that two users are similar because
they prefer one genre in common, while each also likes some genres that
the other doesn’t care for.
