Database Reference
In-Depth Information
In [Ziv04] and [Pap10] the specification of the aggregate groups
G
β
was
performed using AMG methods in combination with RL-specific extensions.
In contrast to this general algebraic case, in most cases product master data is
available for recommendation engines, including their assignment to categories.
This should be used for the construction of the hierarchies, since it contains
important additional information. Thus, there are two sources for specification of
hierarchies for REs:
1. Product hierarchies such as shop taxonomy or product groups
2. Product attributes such as manufacturer, brand, or color
The use of 1. seems obvious; however, in most cases it requires comprehensive
preprocessing so that the pseudo-inverse (
6.12
) can be calculated. In the case of 2.,
this is automatically guaranteed, since every product can be assigned only to one
parent state. Notice that only two-grid hierarchies can be constructed from 2., but in
most cases this is sufficient.
6.2.2 The Model-Based Case: AMG
In the following, we shall investigate and discuss the algebraic multigrid method for
proposed by Bertsekas and Castanon in [BC89]. Without loss of generality, we
shall hold back on the 2-level method, that is,
l ¼
0 is the fine grid and
l ¼
1 the
coarse grid.
Let
L ¼ I
1
be the aggregation prolongator according to (
6.11
). Moreover, let
R
be the restriction operator given by
1
L
T
W
,
W
:¼ L
T
WL
1
R
:¼
diag
ðÞ
,
ð
6
:
14
Þ
n
is (component-
wise) positive and sums to 1. The matrix
R
is also referred to as the
Moore-Penrose
inverse of L with respect to the w-weighted inner product
.
We now consider the 2-level-mutligrid procedure according to Algorithm 6.1 with
a single smoothing step, that is,
(of which (
6.12
) is the special case where
W ¼ I),
where
w
∈
R
ν
1
¼ ν
2
¼
1, by means of the Richardson iteration
x
:¼ IT x
,
y
ð
Þ :¼ x þ y Ax
:
As we carry out only one smoothing step and
y
is taken to be the residual
Ax ^
y
,
the smoothing simplifies to
x
:¼ x þ Ax ^
ð
y
Þ Ax ¼ x ^
y
:
