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associated session vector
b
. This approach basically corresponds to classical
CF, with the only difference that for the storage and evaluation of user profiles
instead of the
m
-dimensional initial space of our products, now the synthesized
k
-dimensional feature space is used. This reduces complexity on the one hand, and
at the same time a generalization of profiles with the associated increase of quality
takes place. Nevertheless, this approach is still complicated and mathematically
difficult to handle.
A better approach is based on projections. Therefore we first want to compute
the incremental ansatz (
8.17
) in a more efficient way. It can be shown that the
following holds:
U
k
U
k
AV
k
V
k
¼ U
k
S
k
V
k
¼ A
k
:
ð
8
:
18
Þ
Because of the elementary relations
V
k
V
k
¼ I
and
U
k
U
k
¼ I
, it follows from
(
8.5
) and (
8.4
) that both
V
k
V
k
as well as
U
k
U
k
are orthoprojectors into the space of
their corresponding singular vector bases. Consequently, the rank-
k
SVD can be
represented as a concatenated projection into the spaces of left and right singular
vector bases.
The projection (
8.18
) can be accomplished even easier.
Proposition 8.6
The following properties hold:
A
k
¼ AV
k
V
k
¼ U
k
U
k
A
Proof
For
A
k
¼ AV
k
V
k
, we just need to replace
AV
k
by
U
k
S
k
according to (
8.11
):
AV
k
V
k
¼ U
k
S
k
V
k
¼ A
k
:
The deduction of
A
k
¼ U
k
U
k
A
is again based on (
8.11
), in conjunction with
(
8.10
) for the decomposition of the Gram matrix by the right singular values, which
constitute its eigenvectors:
U
k
U
k
A ¼ U
k
S
k
V
k
A
T
A ¼ U
k
S
k
V
k
VS
2
V ¼ U
k
S
k
V
k
¼ A
k
□
From Proposition 8.6, it follows that by means of the
right-projection approx-
imation, A
k
can be computed solely via the right singular values:
A
k
¼ AV
k
V
k
:
ð
8
:
19
Þ
Similarly, we can apply the
left-projection approximation
to compute
A
k
only
via the left singular values:
A
k
¼ U
k
U
k
A
:
ð
8
:
20
Þ
