Database Reference
In-Depth Information
Table 8.1 Example of a
session matrix of a web shop
Session 1 Session 2 Session 3 Session 4
Product 1 0
1
10
5
Product 2 1
5
1
1
Product 3 0
5
0
1
statistical models, however, entails some major computational impediments,
including intractable integrals and non-convex optimization, making a realtime
implementation very difficult. The alternative approach consists in treating all
variables as observed by assigning the value 0 to the unknown ratings. Although
this may appear somewhat helter-skelter at the first glance, it may be rationalized
by considering
not visiting a product
as a transaction corresponding to the lowest
possible rating. Assuming, furthermore, that many of the zero entries are due to
noise rather than intrinsic, we may put the approach on a sound footing. We will
return to this discussion in Sect.
8.5
.
Now we consider a matrix of rewards
A
n
p
n
s
with
n
s
being the current
number of sessions and
n
p
the number of different products over all sessions
observed so far. Neither the order of sessions nor the order of products within the
sessions is taken into account.
∈ R
Example 8.1
As an example, we consider a web shop with 3 products and 4
sessions, i.e.,
n
p
¼
3 and
n
s
¼
4. The session values are displayed in Table
8.1
.
In terms of the reward assignment described above, this means, e.g., for session
2, product 1 has been clicked, whereas products 2 and 3 have moreover been added
to the basket. In session 3, product 1 has been purchased, product 2 has been
clicked, and product 3 has been skipped.
■
Mathematically, the matrix factorization problems arising in CF are of the form
rn
s
fA
ð
;
XY
Þ:
ð
8
:
1
Þ
min
,
n
p
r
X
C
1
R
Y
C
2
R
∈
∈
The rank
r
is usually chosen to be considerably smaller than
n
p
. The function
f
is referred to as the cost function of the factorization and, more often than not,
is chosen to be a metric. It stipulates a notion of quality of a factorization.
The sets
C
1
,
C
2
determine the parameter space. In terms of our signal processing
metaphor, the factor
X
characterizes the source, which is restricted to be a
low-dimensional subspace, and the columns
Y
are the intrinsic low-dimensional
parameter vectors determining the signals given by the corresponding columns
of
A
.
To put it even simpler, we approximate the matrix
A
by the product of two
smaller matrices
X
and
Y
. The cost function stipulates a notion of “closeness,” i.e.,
distance, of two matrices. Since the rank
r
is typically much smaller than
n
p
and
n
s
,
the representation in terms of
X
and
Y
is much more compact than an explicit
representation of the entries of
A
.
