Database Reference
In-Depth Information
computation of each possible recommendation requires the computation of all
similarities to all sessions in
N(s, p
). Hence, classical CF can be employed for
small problems only. Due to the lack of generalization,
the quality of the
predictions, too, is amendable.
We finish the short introduction to CF with some technical remarks. Instead of
considering similarities of sessions, we can also look for similarity of products. Let
N(s
) be the set of all products
q
of a given session
s
. Now the predicted reward value
a
sp
of a product
p
not observed in the session is computed as follows:
X
s
pq
a
sq
b
sq
q
∈
NðÞ
X
a
sp
¼ b
sp
þ
^
,
s
pq
q
NðÞ
∈
where
b
sp
is the baseline prediction for
a
sp
(e.g., the mean value of the rewards of
the product over all sessions) and
s
pq
a measure of similarity between the products
p
and
q
using the same similarity measures.
By only considering similarity of products
s
pq
for recommendations, especially
applying the cosine similarity measure, we arrive at the popular
item-to-item
collaborative filtering
[LSY03]. That is, for a product
p
, we simply recommend
the products
q
of maximum similarity values
s
pq
. This simple static type of
recommendations has proved to be very robust and useful in practice (see also
discussion in Sect.
1.6
).
As we already mentioned, different similarity measures can be used for
s
pq
; they
can all be viewed as variants on the inner product. The cosine measure between two
vectors
x
and
y
of length
n
is defined as
X
n
x
i
y
i
ðÞ¼
<
>
kk kk
¼
x
,
y
i¼
1
cos
x
;
s
X
s
X
:
n
n
x
i
y
i
i¼
1
i¼
1
For similarity of products
s
pq
,
x
can be considered as binary vector of the
occurrence of product
p
over all
n
sessions and, similarly,
y
as binary vector of
the occurrence of product
q.
Thus, we can identify
x
with
p
and
y
with
q
. This leads
to an interesting observation.
Since the components
x
i
and
y
i
are binary values, i.e., 0 or 1, by introducing the
support supp
ðÞ¼
X
n
x
i
, we can rewrite the cosine measure as
i¼
1
supp
x ^ y
ð
Þ
supp
x ^ y
ð
Þ
supp
x
^ ð Þ
supp
ðÞ
supp
y
^ ð Þ
supp
ðÞ
¼ e
2
cos
x
ðÞ
;
¼
¼
p
xy
e
p
yx
:
supp
ðÞ
supp
ðÞ
